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2.12: Converse, Inverse, and Contrapositive Statements

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Conditional statements drawn from an if-then statement.

Converse, Inverse, and Contrapositive

Consider the statement: If the weather is nice, then I’ll wash the car. We can rewrite this statement using letters to represent the hypothesis and conclusion.

\(p=the\: weather \:is \:nice \qquad q=I'll \:wash \:the \:car\)

Now the statement is: if \(p\), then \(q\), which can also be written as \(p\rightarrow q\).

We can also make the negations, or “nots,” of \(p\) and \(q\). The symbolic version of "not p" is \(\sim p.

\(\sim p=the \:weather \:is \:not \:nice \qquad \sim q=I \:won't \:wash \:the \:car\)

Using these “nots” and switching the order of \(p\) and \(q\), we can create three new statements.

\(Converse \qquad q\rightarrow p \qquad \underbrace{If\: I\: wash\: the\: car}_\text{q}, \underbrace{then\: the \:weather \:is \: nice}_\text{p}\).

\(Inverse \qquad \sim p\rightarrow \sim q \qquad \underbrace{If\: the\: weather\: is \:not \:nice}_\text{p}, \underbrace{\:then \:I \:won't \:wash \:the \:car}_\text{q}\).

\(Contrapositive \qquad \sim q\rightarrow \sim p \qquad \underbrace{If\: I \:don't \:wash \:the \:car}_\text{q}, \underbrace{then the weather is not nice}_\text{p}\).

If the “if-then” statement is true, then the contrapositive is also true. The contrapositive is logically equivalent to the original statement. The converse and inverse may or may not be true. When the original statement and converse are both true then the statement is a biconditional statement . In other words, if \(p\rightarrow q\) is true and \(q\rightarrow p\) is true, then \(p \leftrightarrow q\) (said “\(p\) if and only if \(q\)”).

What if you were given a conditional statement like "If I walk to school, then I will be late"? How could you rearrange and/or negate this statement to form new conditional statements?

Example \(\PageIndex{1}\)

If \(n>2\), then \(n^{2}>4\).

Find the converse, inverse, and contrapositive. Determine if each resulting statement is true or false. If it is false, find a counterexample.

The original statement is true.

\(\underline{Converse}\): If \(n^{2}>4\), then \(n>2\).

False. If \(n^{2}=9\), \(n=−3\: or \: 3\). \((−3)^{2}=9\)

\(\underline{Inverse}\): If \(n\leq 2\), then \(n^{2}\leq 4\).

False. If \(n=−3\), then \(n^{2}=9\).

\(\underline{Contrapositive}\): If \(n^{2}\leq 4\), then \(n\leq 2\).

True. The only \(n^{2}\leq 4\) is 1 or 4. \(\sqrt{1}=\pm 1\) and\(\sqrt{4}=\pm 2\), which are both less than or equal to 2.

Example \(\PageIndex{2}\)

If I am at Disneyland, then I am in California.

\(\underline{Converse}\): If I am in California, then I am at Disneyland.

False. I could be in San Francisco.

\(\underline{Inverse}\): If I am not at Disneyland, then I am not in California.

False. Again, I could be in San Francisco.

\(\underline{Contrapositive}\): If I am not in California, then I am not at Disneyland.

True. If I am not in the state, I couldn't be at Disneyland.

Notice for the converse and inverse we can use the same counterexample.

Example \(\PageIndex{3}\)

Rewrite as a biconditional statement: Any two points are collinear.

This statement can be rewritten as:

Two points are on the same line if and only if they are collinear. Replace the “if-then” with “if and only if” in the middle of the statement.

Example \(\PageIndex{4}\)

Any two points are collinear.

First, change the statement into an “if-then” statement:

If two points are on the same line, then they are collinear.

\(\underline{Converse}\): If two points are collinear, then they are on the same line. True.

\(\underline{Inverse}\): If two points are not on the same line, then they are not collinear. True.

\(\underline{Contrapositive}\): If two points are not collinear, then they do not lie on the same line. True.

Example \(\PageIndex{5}\)

The following is a true statement:

\(m\angle ABC>90^{\circ}\) if and only if \(\angle ABC\) is an obtuse angle.

Determine the two true statements within this biconditional.

Statement 1: If \(m\angle ABC>90^{\circ}\), then \(\angle ABC\) is an obtuse angle.

Statement 2: If \(\angle ABC\) is an obtuse angle, then \(m\angle ABC>90^{\circ}\).

For questions 1-4, use the statement:

If \(AB=5\) and \(BC=5\), then \(B\) is the midpoint of \(\overline{AC}\).

• Is this a true statement? If not, what is a counterexample?
• Find the converse of this statement. Is it true?
• Find the inverse of this statement. Is it true?
• Find the contrapositive of this statement. Which statement is it the same as?

Find the converse of each true if-then statement. If the converse is true, write the biconditional statement.

• An acute angle is less than \(90^{\circ}\).
• If you are at the beach, then you are sun burnt.
• If \(x>4\), then \(x+3>7\).

For questions 8-10, determine the two true conditional statements from the given biconditional statements.

• A U.S. citizen can vote if and only if he or she is 18 or more years old.
• A whole number is prime if and only if its factors are 1 and itself.
• \(2x=18\) if and only if \(x=9\).

Review (Answers)

To see the Review answers, open this PDF file and look for section 2.4.

Additional Resources

Interactive Element

Video: Converse, Inverse and Contrapositive of a Conditional Statement Principles - Basic

Activities: Converse, Inverse, and Contrapositive Discussion Questions

Study Aids: Conditional Statements Study Guide

Practice: Converse, Inverse, and Contrapositive Statements

Real World: Converse Inverse Contrapositive

Converse Statement

A statement that is of the form "If p then q" is a conditional statement. Here 'p' refers to 'hypotheses' and 'q' refers to 'conclusion'.

For example, "If Cliff is thirsty, then she drinks water."  This is a conditional statement. It is also called an implication . The converse statement is " If Cliff drinks water then she is thirsty". 

A converse statement is the opposite of a conditional statement. It is to be noted that not always the converse of a conditional statement is true. 

For example, in geometry , "If a closed shape has four sides then it is a square" is a conditional statement, The truthfulness of a converse statement depends on the truth of hypotheses of the conditional statement.

In this mini-lesson, we will learn about the converse statement, how inverse and contrapositive are obtained from a conditional statement, converse statement definition, converse statement geometry, and converse statement symbol.

Lesson Plan

What is converse statement.

A statement obtained by reversing the hypothesis and conclusion of a conditional statement is called a converse statement.

Explanation

Let us understand the terms "hypothesis" and "conclusion."

A statement which is of the form of "if p then q" is a conditional statement, where 'p' is called hypothesis and 'q' is called the conclusion.

A converse statement is gotten by exchanging the positions of 'p' and 'q' in the given condition.

For example,

"If Cliff is thirsty, then she drinks water" is a condition.

The converse statement is "If Cliff drinks water, then she is thirsty."

The hypothesis 'p' and conclusion 'q' interchange their places in a converse statement.

What Is Inverse Statement?

A statement obtained by negating the hypothesis and conclusion of a conditional statement.

An inverse statement changes the "if p then q" statement to the form of  "if not p then not q."

"If John has time, then he works out in the gym."

The inverse statement is "If John does not have time, then he does not work out in the gym." 

What Is Contrapositive Statement?

A statement obtained by exchanging the hypothesis and conclusion of an inverse statement. 

A contrapositive statement changes "if not p then not q" to "if not q to then, not p."

• The converse of the conditional statement is “If  Q  then  P .”
• The contrapositive of the conditional statement is “If not  Q  then not  P .”
• The inverse of the conditional statement is “If not  P  then not  Q .”

If it is a holiday, then I will wake up late. - Conditional statement

If it is not a holiday, then I will not wake up late. - Inverse statement

If I am not waking up late, then it is not a holiday. - Contrapositive statement

What Is a Conditional Statement? 

A conditional statement is a statement in the form of "if p then q," where 'p' and 'q' are called a hypothesis and conclusion.

A conditional statement defines that if the hypothesis is true then the conclusion is true.

"If we have to to travel for a long distance, then we have to take a taxi" is a conditional statement.

Converse of a Conditional Statement

To get the converse of a conditional statement, interchange the places of hypothesis and conclusion.

If you eat a lot of vegetables, then you will be healthy. - Conditional statement

If you are healthy, then you eat a lot of vegetables. - Converse of Conditional statement

Inverse of Conditional Statement

To get the inverse of a conditional statement, we negate both the hypothesis and conclusion.

If you read books, then you will gain knowledge. - Conditional statement

If you do not read books, then you will not gain knowledge. -Inverse of conditional statement

Contrapositive of Conditional Statement

To get the contrapositive of a conditional statement, we negate the hypothesis and conclusion and exchange their position.

Emily's dad watches a movie if he has time. - Conditional statement

If Emily's dad does not have time, then he does not watch a movie. - Contrapositive of a conditional statement

• In a conditional statement "if p then q," 'p' is called the hypothesis and 'q' is called the conclusion.
• There can be three related logical statements for a conditional statement.

Solved Examples

Write the converse, inverse, and contrapositive statement of the following conditional statement.

If you win the race then you will get a prize.

The conditional statement given is "If you win the race then you will get a prize." 

It is of the form "If p then q".

Here 'p' is the hypothesis and 'q' is the conclusion.

From the given inverse statement, write down its conditional and contrapositive statements.

If there is no accomodation in the hotel, then we are not going on a vacation.

The inverse statement given is "If there is no accomodation in the hotel, then we are not going on a vacation."

It is of the form "If not p then not q"

Write the converse, inverse, and contrapositive statement for the following conditional statement. 

If you study well then you will pass the exam. Solution

Given statement is - If you study well then you will pass the exam.

• If 2a + 3 <  10, then a = 3. Write the converse, inverse, and contrapositive statements and verify their truthfulness.

Interactive Questions 

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

Let's Summarize

The mini-lesson targeted the fascinating concept of converse statement. Hope you enjoyed learning! Now you can easily find the converse, inverse, and contrapositive of any conditional statement you are given!

At  Cuemath , our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

Frequently Asked Questions (FAQs)

1. what is the negation of a statement.

A statement that conveys the opposite meaning of a statement is called its negation.

2. How do you write a converse statement?

A statement formed by interchanging the hypothesis and conclusion of a statement is its converse.

• Live one on one classroom and doubt clearing
• Practice worksheets in and after class for conceptual clarity
• Personalized curriculum to keep up with school

Converse, Inverse, and Contrapositive

As your English teacher would say, good writers vary their sentence structure. The same is true of conditional statements: after a while, the If-Then formula becomes a real snoozefest. Some ways to mix it up are: "All things satisfying hypothesis are conclusion " and " Conclusion whenever hypothesis ."

However, mathematicians can be drier than the Sahara desert: they tend to write conditional statements as a formula p → q , where p is the hypothesis and q the conclusion. In fact, the old saying, "Mind your p 's and q 's," has its origins in this sort of mathematical logic.

Sample Problem

Identify p and q in the following statements, translating them into p → q form.

(A) If it rains outside, then flowers will grow tomorrow. (B) I cut off a finger whenever I peel rutabagas. (C) All dogs go to heaven.

For (A), p = "it rains outside" and q = "flowers will grow tomorrow."

In (B), we may rewrite the statement as "If I peel rutabagas, then I cut off a finger," telling us that p = "I peel rutabagas" and q = "I cut off a finger."

Finally, we may rewrite (C) as "If it is a dog, then it will go to heaven," yielding p = "it is a dog" and q = "it will go to heaven."

The hypothesis and conclusion play very different roles in conditional statements. Duh. In other words, p → q and q → p mean very different things. It's kind of like subtraction: 5 – 3 gives a different answer than 3 – 5. To highlight this distinction, mathematicians have given a special name to the statement q → p : it's called the converse of p → q .

No, not those Converse.

Write the converse of the statement, "If something is a watermelon, then it has seeds."

We want to switch the hypothesis and the conclusion, which will give us: "If something has seeds, then it is a watermelon." Of course, this converse is obviously false, since apples, cucumbers, and sunflowers all have seeds and are not watermelons. At least not during their day jobs.

There are some other special ways of modifying implications. For example, if you negate (that means stick a "not" in front of) both the hypothesis and conclusion, you get the inverse : in symbols, not p → not q is the inverse of p → q . Sometimes mathematicians like to be even more brief than this, so they'll abbreviate "not" with the symbol "~". So we can also write the inverse of p → q as ~ p → ~ q .

Finally, if you negate everything and flip p and q (taking the inverse of the converse, if you're fond of wordplay) then you get the contrapositive . Again in symbols, the contrapositive of p → q is the statement not q → not p , or ~ q → ~ p . Fancy.

What is the inverse of the statement "All mirrors are shiny?" What is its contrapositive?

If we abbreviate the first statement as mirror → shiny, then the inverse would be not mirror → not shiny and the contrapositive would be not shiny → not mirror. Written in English, the inverse is, "If it is not a mirror, then it is not shiny," while the contrapositive is, "If it is not shiny, then it is not a mirror."

While we've seen that it's possible for a statement to be true while its converse is false, it turns out that the contrapositive is better behaved. Whenever a conditional statement is true, its contrapositive is also true and vice versa. Similarly, a statement's converse and its inverse are always either both true or both false. (Note that the inverse is the contrapositive of the converse. Can you show that?)

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W hy's T his F unny?

Converse, Inverse, and Contrapositive

This is the third post in a series on logic, with a focus on how it is expressed in English. We’ve looked at basic ideas of translating between English and logical symbols, and in particular at negation (stating the opposite). Now we are ready to consider how to change a given statement into one of three related statements.

A conditional statement and its converse

We’ll start with a question from 1999 that introduces the concepts:

Ricky has been asked to break down the statement, “A number divisible by 2 is divisible by 4,” into its component parts, and then rearrange them to find the converse of the statement. I took the question:

We commonly write such a statement symbolically as “\(p\rightarrow q\)“, where the hypothesis is p and the conclusion is q . I rewrote each part slightly to allow it to exist outside of the sentence, naming the number N to avoid needing pronouns. What was important was to rewrite the statement in if/then form.

The converse of this statement swaps the hypothesis and conclusion, making “\(q\rightarrow p\)“:

Ricky was asked to decide whether the converse is true or not, and then prove it, whichever way it goes. This part goes beyond mere logic and enters the realm of “number theory”; but commonly this sort of question is first asked in cases where the proof is not too hard, which is the case here.

To show that a statement is not always true, we only need to find an example for which it is false. In this case, an easy example is 2, or we could use 6, or 102, or whatever we like.

But the question was about the converse:

I didn’t give a proof, in part because Ricky needed to think about that for himself, but also because I didn’t know what level of proof Ricky is expected to handle. One approach is to see that any multiple of 4 can be written as 4 k for some integer k ; but that can be written as 2(2 k ), which is clearly a multiple of 2.

Converse, inverse, and contrapositive

Now we can review the meanings of all three terms, in this 1999 question, which again uses an example from basic number theory:

Doctor Kate could have asked Hollye  for  her  answers to part A, to make sure she understands that part; but she chose to provide them:

It’s important to identify the parts of a conditional statement (if p then q ); and since two of the new statements require negations, that also might as well be done early. Notice that the negation of “is even” could have been written as “is not even”, but since every number (integer) is either odd or even, writing “is odd” is cleaner. Also, the negation of “both are even” is “at least one is not even”; this is an application of De Morgan’s law, or can be seen by considering that if it is not true that both are even, then there must be one that is not even. These ideas were discussed last time.

Now here are the new statements:

We saw the converse above; there we just swap p and q . The inverse keeps each part in place, but negates it. The contrapositive both swaps and negates the parts.

So now we know that the contrapositive, “If either m or n is odd, then m + n is odd,” is false, because there is at least one case, 3 and 7, where the hypothesis is true but the conclusion is false.

That’s the essence of a counterexample.

Doctor Kate continued, showing a way to prove that B and C (the converse and inverse) are both false. You can read that on your own, since my goal here is just to look at the logic. (We’ll have a series on proofs some time in the future.)

Rewriting the statement

Continuing, here is a similar question, where statements must first be written in conditional form:

The second statement is straightforward, but the others need thought. Doctor Achilles first defined the three forms, as we’ve already seen, and then dealt with the first case:

Thus, “all” (the universal quantifier) translates directly to a conditional. The answer, left for Hana to do, will be:

• Converse: “If x is a quadrilateral, then x is a square”; i.e. “Any quadrilateral is a square.”
• Inverse: “If x is not a square, then x is not a quadrilateral”; i.e. “Anything that is not a square is not a quadrilateral.”
• Contrapositive: “If x is not a quadrilateral, then x is not a square”; i.e. “Anything that is not a quadrilateral is not a square.”

The original statement, and the contrapositive, are true, because a square is a kind of quadrilateral; the converse and inverse are false, and a counterexample would be an oblong rectangle, which is not a square but is a quadrilateral.

The questions so far, where they dealt with truth at all, only asked about specific examples. Our last two questions will look more broadly at when these statements are equivalent.

Which can I use in a proof?

Consider this question, from 2002:

If we know a statement is true, can we conclude that the inverse is true? Doctor TWE answered with a counterexample:

Here we are using logic to talk about logic: The statement “For all p and q , \((p\rightarrow q)\rightarrow(\lnot p\rightarrow\lnot q)\)” is false! Sometimes both original and inverse are true, but we can’t conclude the latter from the former.

Giving one example where the contrapositive is true does not prove that it is always equivalent; we’ll prove it below.

In fact, the converse and inverse turn out to be equivalent to one another, though not to the original.

Why is the contrapositive equivalent?

Let’s look at one more, from 2003:

The opening statement describes the contrapositive as the inverse of the converse. What that means is this: Suppose we start with “\(p\rightarrow q\)“. Its converse is “\(q\rightarrow p\)” (swapping the order), and the inverse of that is “\(\lnot q\rightarrow\lnot p\)” (negating each part). This is the contrapositive. In the example, the converse of “If I like cats, then I have cats” is “If I have cats, then I like cats”, and the inverse of that is “If I don’t have cats, then I don’t like cats”, which is the contrapositive.

Doctor Achilles, perhaps misreading the question, answered the bigger question: Which of these are true?

In effect, he has made a truth table:

If you are unconvinced by any of the reasoning, see  Why, in Logic, Does False Imply Anything? .

So the truth table for the contrapositive is that same as for the original; this is what we mean when we say that two statements are logically equivalent .

We can instead just think through the example:

Which is more convincing? That depends upon you.

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This wiki is incomplete.

Switching the hypothesis and conclusion of a conditional statement gives a converse. For example, the converse of "If it is raining then the grass is wet" is "If the grass is wet then it is raining."

Note: As in the example, a proposition may be true but have a false converse.

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Converse, Inverse, and Contrapositive

Hi, and welcome to this video on mathematical statements! Today, we’ll be exploring the logic that appears in the language of math. Specifically, we will learn how to interpret a math statement to create what are known as converse, inverse, and contrapositive statements. These, along with some reasoning skills, allow us to make sense of problems presented in math. Let’s get started!

Let’s first take a look at a basic statement , which can be either true or false, but never both. For example, a declarative statement pronounces a fact, like “the Sun is hot.” We know this is a statement because the Sun cannot be both hot and not hot at the same time. This declarative statement could also be referred to as a proposition .

Two independent statements can be related to each other in a logic structure called a conditional statement . The first statement is presented with “if,” and is referred to as the hypothesis . The second statement is linked with “then”, and is known as the conclusion . The notation associated with conditional statements typically uses the variable \(p\) for the hypothesis statement, and \(q\) for the conclusion.

  In words, this would be read as, “If \(p\), then \(q\).”

When the hypothesis and conclusion are identified in a statement, three other statements can be derived:

• The contrapositive statement is a combination of the previous two. The positions of \(p\) and \(q\) of the original statement are switched, and then the opposite of each is considered: \(\sim q \rightarrow \sim p\) (if not \(q\), then not \(p\)).

An example will help to make sense of this new terminology and notation. Let’s start with a conditional statement and turn it into our three other statements.

  The first step is to identify the hypothesis and conclusion statements. Conditional statements make this pretty easy, as the hypothesis follows if and the conclusion follows then . The hypothesis is it is raining and the conclusion is grass is wet .

  Now we can use the definitions that we introduced earlier to create the three other statements.

• Our contrapositive statement would be: “If the grass is NOT wet, then it is NOT raining.”

You may be wondering why we would want to go through the trouble of rearranging and considering the “opposite” of the hypothesis and conclusion statements. How is this helpful? The key is in the relationship between the statements. If we know that a statement is true (or false), then we can assume that another is also true (or false). The statements that are related in this way are considered logically equivalent .

For example, consider the statement, “If it is raining, then the grass is wet” to be TRUE. Then you can assume that the contrapositive statement, “If the grass is NOT wet, then it is NOT raining” is also TRUE.

Likewise, the converse statement, “If the grass is wet, then it is raining” is logically equivalent to the inverse statement, “If it is NOT raining, then the grass is NOT wet.”

These relationships are particularly helpful in math courses when you are asked to prove theorems based on definitions that are already known. Much of that work is beyond the scope of this video, but the following examples will help to illustrate the relationships of logically equivalent statements.

Here is a typical example of a TRUE statement that would be made in a geometry class based on the definition of congruent angles :

  As you can see, this is not a conditional statement, but we can rewrite it in the “if-then” structure to identify the hypothesis and conclusion statements as follows:

  Now we have a hypothesis and a conclusion.

  Because the conditional statement and the contrapositive are logically equivalent, we can assume the following to be TRUE:

  It follows that the converse statement, “If two angles are congruent, then the two angles have the same measure,” is logically equivalent to the inverse statement, “If two angles do NOT have the same measure, then they are NOT congruent.”

Here is another example of a TRUE statement:

  The conditional statement would be “If a figure is a square, then it is a rectangle,” which gives us our hypothesis and conclusion.

  Because the contrapositive statement is logically equivalent, we can assume that “If the figure is NOT a rectangle, then the figure is NOT a square” is also a TRUE statement.

However, the converse statement can be disproved.

As can be seen in the diagram above, squares are a type of rectangle and a rectangle is a type of polygon . However, a square is a special type of rectangle that has four sides of equal length. Not all rectangles have four equal sides like a square, so our converse statement is FALSE.

Accordingly, the inverse statement is also FALSE because they are logically equivalent:

  In summary, the original statement is logically equivalent to the contrapositive, and the converse statement is logically equivalent to the inverse.

That is a lot to take in! Let’s end this video with an example for you to process how to analyze a statement to write the converse, inverse, and contrapositive statements.

For this exercise, don’t worry about whether the statements are true or false. The statement is:

  Now, pause the video and see if you can figure out the converse, inverse, and contrapositive statements. Remember, it helps to first turn our original statement into a conditional statement so you know the hypothesis and conclusion.

Okay, let’s see if you figured it out!

The conditional statement would be: “If all figures are four-sided planes, then figures are rectangles.” This gave us our hypothesis and conclusion.

  Here are the converse, inverse, and contrapositive statements based on the hypothesis and conclusion:

Converse : “If figures are rectangles, then figures are all four-sided planes.”

Inverse : “If figures are NOT all four-sided planes, then they are NOT rectangles.”

Contrapositive : “If figures are NOT rectangles, then the figures are NOT all four-sided planes.”

That’s all for this review! Thanks for watching, and happy studying!

Converse, Inverse, and Contrapositive Practice Questions

  Which statement is NOT considered a conditional statement?

If you mow the lawn, then I will pay you for your hard work.

If you pay your power bill, then you will have electricity.

If you do not buy firewood, then you will be cold.

The sun is shining, because it is summer.

Conditional statements are also considered “If-Then” statements. An “If-Then” statement consists of a hypothesis (if) and a conclusion (then). For example, If it is snowing, then it is cold. The logic structure of conditional statements is helpful for deriving converse, inverse, and contrapositive statements.

  What is the inverse statement of the following conditional statement? If it is snowing, then it is cold.

If it is not snowing, then it is cold.

If it is not snowing, then it is not cold.

If it is cold, then it might be snowing.

If it is cold, then it is not warm.

An inverse statement assumes the opposite of each of the original statements. The opposite of “If it is snowing” would be “If it is not snowing.” The opposite of “then it is cold” would be “then it is not cold.”

  What is the contrapositive statement for the following conditional statement? If it is a triangle, then it is a polygon.

If it is not a triangle, then it is not a polygon.

If it is not a polygon, then it is not a triangle.

If it is a triangle, then it is a polygon.

If it is a polygon, it is a triangle.

A contrapositive statement occurs when you switch the hypothesis and the conclusion in a statement, and negate both statements. In this example, when we switch the hypothesis and the conclusion, and negate both, the result is: If it is not a polygon, then it is not a triangle.

  Identify p (hypothesis) and q (conclusion) in the following conditional statement. If a figure is a triangle, then it has three angles.

q : If a figure is a triangle p : Then it has congruent angles

p : A figure is a polygon q : Can have three angles

p : If a figure is not a triangle q : Then it does not have three angles

p : If a figure is a triangle q : Then it has three angles

The hypothesis ( p ) of a conditional statement is the “if” portion. The conclusion ( q ) of a conditional statement is the “then” portion.

  Which item shows the math statements matched with the correct logic symbols?

Conditional Statement: q → p Converse: q → p Inverse: ~ p → ~ q Contrapositive: ~ p → ~ q

Conditional Statement: p → q Converse: q → p Inverse: ~ p → ~ q Contrapositive: ~ q → ~ p

Conditional Statement: r → t Converse: q → p Inverse: ~ t → ~ r Contrapositive: ~ q → ~ p

Conditional Statement: p → q Converse: q → p Inverse: p → q Contrapositive: ~ h → ~ c

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2.5 Equivalent Statements

Learning objectives.

After completing this section, you should be able to:

• Determine whether two statements are logically equivalent using a truth table.
• Compose the converse, inverse, and contrapositive of a conditional statement

Have you ever had a conversation with or sent a note to someone, only to have them misunderstand what you intended to convey? The way you choose to express your ideas can be as, or even more, important than what you are saying. If your goal is to convince someone that what you are saying is correct, you will not want to alienate them by choosing your words poorly.

Logical arguments can be stated in many different ways that still ultimately result in the same valid conclusion. Part of the art of constructing a persuasive argument is knowing how to arrange the facts and conclusion to elicit the desired response from the intended audience.

In this section, you will learn how to determine whether two statements are logically equivalent using truth tables, and then you will apply this knowledge to compose logically equivalent forms of the conditional statement. Developing this skill will provide the additional skills and knowledge needed to construct well-reasoned, persuasive arguments that can be customized to address specific audiences.

An alternate way to think about logical equivalence is that the truth values have to match. That is, whenever p p is true, q q is also true, and whenever p p is false, q q is also false.

Determine Logical Equivalence

Two statements, p p and q q , are logically equivalent when p ↔ q p ↔ q is a valid argument, or when the last column of the truth table consists of only true values. When a logical statement is always true, it is known as a tautology . To determine whether two statements p p and q q are logically equivalent, construct a truth table for p ↔ q p ↔ q and determine whether it valid. If the last column is all true, the argument is a tautology, it is valid, and p p is logically equivalent to q q ; otherwise, p p is not logically equivalent to q q .

Example 2.22

Determining logical equivalence with a truth table.

Create a truth table to determine whether the following compound statements are logically equivalent.

• p → q ; p → q ; ~ p → ~ q ~ p → ~ q
• p → q ; p → q ; ~ p ∨ q ~ p ∨ q

Because the last column it not all true, the biconditional is not valid and the statement p → q p → q is not logically equivalent to the statement ~ p → ~ q ~ p → ~ q .

Because the last column is true for every entry, the biconditional is valid and the statement p → q p → q is logically equivalent to the statement ~ p ∨ q ~ p ∨ q . Symbolically, p → q ≡ ~ p ∨ q . p → q ≡ ~ p ∨ q .

Your Turn 2.22

Compose the converse, inverse, and contrapositive of a conditional statement.

The converse , inverse , and contrapositive are variations of the conditional statement, p → q . p → q .

• The converse is if q q then p p , and it is formed by interchanging the hypothesis and the conclusion. The converse is logically equivalent to the inverse.
• The inverse is if ~ p ~ p then ~ q ~ q , and it is formed by negating both the hypothesis and the conclusion. The inverse is logically equivalent to the converse.
• The contrapositive is if ~ q ~ q then ~ p ~ p , and it is formed by interchanging and negating both the hypothesis and the conclusion. The contrapositive is logically equivalent to the conditional.

The table below shows how these variations are presented symbolically.

Example 2.23

Writing the converse, inverse, and contrapositive of a conditional statement.

Use the statements, p p : Harry is a wizard and q q : Hermione is a witch, to write the following statements:

• Write the conditional statement, p → q p → q , in words.
• Write the converse statement, q → p q → p , in words.
• Write the inverse statement, ~ p → ~ q ~ p → ~ q , in words.
• Write the contrapositive statement, ~ q → ~ p ~ q → ~ p , in words.
• The conditional statement takes the form, “if p p , then q q ,” so the conditional statement is: “If Harry is a wizard, then Hermione is a witch.” Remember the if … then … words are the connectives that form the conditional statement.
• The converse swaps or interchanges the hypothesis, p p , with the conclusion, q q . It has the form, “if q q , then p p .” So, the converse is: "If Hermione is a witch, then Harry is a wizard."
• To construct the inverse of a statement, negate both the hypothesis and the conclusion. The inverse has the form, “if ~ p ~ p , then ~ q ~ q ,” so the inverse is: "If Harry is not a wizard, then Hermione is not a witch."
• The contrapositive is formed by negating and interchanging both the hypothesis and conclusion. It has the form, “if ~ q ~ q , then ~ p ~ p ," so the contrapositive statement is: "If Hermione is not a witch, then Harry is not a wizard."

Your Turn 2.23

Example 2.24, identifying the converse, inverse, and contrapositive.

Use the conditional statement, “If all dogs bark, then Lassie likes to bark,” to identify the following.

• Write the hypothesis of the conditional statement and label it with a p p .
• Write the conclusion of the conditional statement and label it with a q q .
• Identify the following statement as the converse, inverse, or contrapositive: “If Lassie likes to bark, then all dogs bark.”
• Identify the following statement as the converse, inverse, or contrapositive: “If Lassie does not like to bark, then some dogs do not bark.”
• Which statement is logically equivalent to the conditional statement?
• The hypothesis is the phrase following the if . The answer is p p : All dogs bark. Notice, the word if is not included as part of the hypothesis.
• The conclusion of a conditional statement is the phrase following the then . The word then is not included when stating the conclusion. The answer is: q q : Lassie likes to bark.
• “Lassie likes to bark” is q q and “All dogs bark” is p p . So, “If Lassie likes to bark, then all dogs bark,” has the form “if q q , then p p ,” which is the form of the converse.
• “Lassie does not like to bark” is ~ q ~ q and “Some dogs do not bark” is ~ p ~ p . The statement, “If Lassie does not like to bark, then some dogs do not bark,” has the form “if ~ q ~ q , then ~ p ~ p ,” which is the form of the contrapositive.
• The contrapositive ~ q → ~ p ~ q → ~ p is logically equivalent to the conditional statement p → q . p → q .

Your Turn 2.24

Example 2.25, determining the truth value of the converse, inverse, and contrapositive.

Assume the conditional statement, p → q : p → q : “If Chadwick Boseman was an actor, then Chadwick Boseman did not star in the movie Black Panther ” is false, and use it to answer the following questions.

• Write the converse of the statement in words and determine its truth value.
• Write the inverse of the statement in words and determine its truth value.
• Write the contrapositive of the statement in words and determine its truth value.
• The only way the conditional statement can be false is if the hypothesis, p p : Chadwick Boseman was an actor, is true and the conclusion, q q : Chadwick Boseman did not star in the movie Black Panther , is false. The converse is q → p , q → p , and it is written in words as: “If Chadwick Boseman did not star in the movie Black Panther , then Chadwick Boseman was an actor.” This statement is true, because false → → true is true.
• The inverse has the form “ ~ p → ~ q . ~ p → ~ q . ” The written form is: “If Chadwick Boseman was not an actor, then Chadwick Boseman starred in the movie Black Panther .” Because p p is true, and q q is false, ~ p ~ p is false, and ~ q ~ q is true. This means the inverse is false → → true, which is true. Alternatively, from Question 1, the converse is true, and because the inverse is logically equivalent to the converse it must also be true.
• The contrapositive is logically equivalent to the conditional. Because the conditional is false, the contrapositive is also false. This can be confirmed by looking at the truth values of the contrapositive statement. The contrapositive has the form “ ~ q → ~ p ~ q → ~ p .” Because q q is false and p p is true, ~ q ~ q is true and ~ p ~ p is false. Therefore, ~ q → ~ p ~ q → ~ p is true → → false, which is false. The written form of the contrapositive is “If Chadwick Boseman starred in the movie Black Panther , then Chadwick Boseman was not an actor.”

Your Turn 2.25

Check your understanding, section 2.5 exercises.

• Write the conditional statement p → q in words.
• Write the converse statement q → p in words.
• Write the inverse statement ~ p → ~ q in words.
• Write the contrapositive statement ~ q → ~ p in words.

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Converse, Inverse, Contrapositive

Given an if-then statement "if p , then q ," we can create three related statements:

A conditional statement consists of two parts, a hypothesis in the “if” clause and a conclusion in the “then” clause.  For instance, “If it rains, then they cancel school.”    "It rains" is the hypothesis.   "They cancel school" is the conclusion.

To form the converse of the conditional statement, interchange the hypothesis and the conclusion.       The converse of "If it rains, then they cancel school" is "If they cancel school, then it rains."

To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion.       The inverse of “If it rains, then they cancel school” is “If it does not rain, then they do not cancel school.”

To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement.        The contrapositive of "If it rains, then they cancel school" is "If they do not cancel school, then it does not rain."

If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.

In the above example, since the hypothesis and conclusion are equivalent, all four statements are true. But this will not always be the case!

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2.3: Converse, Inverse, and Contrapositive

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Related to the conditional \(p \rightarrow q\) are three important variations. 

Definition: Converse

\(\displaystyle q \rightarrow p\)

Definition: Inverse

\(\displaystyle \neg p \rightarrow \neg q\)

Definition: Contrapositive

\(\displaystyle \neg q \rightarrow \neg p\)

Theorem \(\PageIndex{1}\): Modus Tollens

A conditional and its contrapositive are equivalent. 

We simply compare the truth tables.

As the two “output” columns are identical, we conclude that the statements are equivalent.

Corollary \(\PageIndex{1}\): Modus Tollens for Inverse and Converse

The inverse and converse of a conditional are equivalent. 

The inverse of the conditional \(p \rightarrow q\) is \(\neg p \rightarrow \neg q\text{.}\) The contrapositive of this new conditional is \(\neg \neg q \rightarrow \neg \neg p\text{,}\) which is equivalent to \(q \rightarrow p\) by double negation.

Warning \(\PageIndex{1}\): Common Mistakes

• Mixing up a conditional and its converse.
• Assuming that a conditional and its converse are equivalent.

Example \(\PageIndex{1}\): Related Conditionals are not All Equivalent

Suppose \(m\) is a fixed but unspecified whole number that is greater than \(2\text{.}\)

Only two of these four statements are true!

Suppose \(f(x)\) is a fixed but unspecified function. 

What Are the Converse, Contrapositive, and Inverse?

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Conditional statements make appearances everywhere. In mathematics or elsewhere, it doesn’t take long to run into something of the form “If P then Q .” Conditional statements are indeed important. What is also important are statements that are related to the original conditional statement by changing the position of P , Q and the negation of a statement. Starting with an original statement, we end up with three new conditional statements that are named the converse, the contrapositive, and the inverse .

Before we define the converse, contrapositive, and inverse of a conditional statement, we need to examine the topic of negation. Every statement in logic is either true or false. The negation of a statement simply involves the insertion of the word “not” at the proper part of the statement. The addition of the word “not” is done so that it changes the truth status of the statement.

It will help to look at an example. The statement “The right triangle is equilateral” has negation “The right triangle is not equilateral.” The negation of “10 is an even number” is the statement “10 is not an even number.” Of course, for this last example, we could use the definition of an odd number and instead say that “10 is an odd number.” We note that the truth of a statement is the opposite of that of the negation.

We will examine this idea in a more abstract setting. When the statement P is true, the statement “not P ” is false. Similarly, if P is false, its negation “not ​ P ” is true. Negations are commonly denoted with a tilde ~. So instead of writing “not P ” we can write ~ P .

Converse, Contrapositive, and Inverse

Now we can define the converse, the contrapositive and the inverse of a conditional statement. We start with the conditional statement “If P then Q .”

• The converse of the conditional statement is “If Q then P .”
• The contrapositive of the conditional statement is “If not Q then not P .”
• The inverse of the conditional statement is “If not P then not Q .”

We will see how these statements work with an example. Suppose we start with the conditional statement “If it rained last night, then the sidewalk is wet.”

• The converse of the conditional statement is “If the sidewalk is wet, then it rained last night.”
• The contrapositive of the conditional statement is “If the sidewalk is not wet, then it did not rain last night.”
• The inverse of the conditional statement is “If it did not rain last night, then the sidewalk is not wet.”

Logical Equivalence

We may wonder why it is important to form these other conditional statements from our initial one. A careful look at the above example reveals something. Suppose that the original statement “If it rained last night, then the sidewalk is wet” is true. Which of the other statements have to be true as well?

• The converse “If the sidewalk is wet, then it rained last night” is not necessarily true. The sidewalk could be wet for other reasons.
• The inverse “If it did not rain last night, then the sidewalk is not wet” is not necessarily true. Again, just because it did not rain does not mean that the sidewalk is not wet.
• The contrapositive “If the sidewalk is not wet, then it did not rain last night” is a true statement.

What we see from this example (and what can be proved mathematically) is that a conditional statement has the same truth value as its contrapositive. We say that these two statements are logically equivalent. We also see that a conditional statement is not logically equivalent to its converse and inverse.

Since a conditional statement and its contrapositive are logically equivalent, we can use this to our advantage when we are proving mathematical theorems. Rather than prove the truth of a conditional statement directly, we can instead use the indirect proof strategy of proving the truth of that statement’s contrapositive. Contrapositive proofs work because if the contrapositive is true, due to logical equivalence, the original conditional statement is also true.

It turns out that even though the converse and inverse are not logically equivalent to the original conditional statement , they are logically equivalent to one another. There is an easy explanation for this. We start with the conditional statement “If Q then P ”. The contrapositive of this statement is “If not P then not Q .” Since the inverse is the contrapositive of the converse, the converse and inverse are logically equivalent.

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Contrapositive and Converse

You may come across different types of statements in mathematical reasoning where some are mathematically acceptable statements and some are not acceptable mathematically. In mathematics, we observe many statements with “if-then” frequently. For example, consider the statement. Contrapositive and converse are specific separate statements composed from a given statement with “if-then”. Before getting into the contrapositive and converse statements, let us recall what are conditional statements. A conditional statement is formed by “if-then” such that it contains two parts namely hypothesis and conclusion. Hypothesis exists in the”if” clause, whereas the conclusion exists in the “then” clause.

What are Contrapositive Statements?

It is easy to understand how to form a contrapositive statement when one knows about the inverse statement. To create the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion. First, form the inverse statement, then interchange the hypothesis and the conclusion to write the conditional statement’s contrapositive.

Click here to know how to write the negation of a statement .

In other words, contrapositive statements can be obtained by adding “not” to both component statements and changing the order for the given conditional statements.

What are Converse Statements?

The converse statements are formed by interchanging the hypothesis and conclusion of given conditional statements.

Thus, we can relate the contrapositive, converse and inverse statements in such a way that the contrapositive is the inverse of a converse statement.

This can be better understood with the help of an example.

Example: Consider the following conditional statement.

If a number is a multiple of 8, then the number is a multiple of 4.

Write the contrapositive and converse of the statement.

Given conditional statement is:

The converse of the above statement is:

If a number is a multiple of 4, then the number is a multiple of 8.

The inverse of the given statement is obtained by taking the negation of components of the statement.

If a number is not a multiple of 8, then the number is not a multiple of 4.

Now, the contrapositive statement is:

If a number is not a multiple of 4, then the number is not a multiple of 8.

All these statements may or may not be true in all the cases. That means, any of these statements could be mathematically incorrect.

Contrapositive vs Converse

The differences between Contrapositive and Converse statements are tabulated below.

We can also construct a truth table for contrapositive and converse statement.

The truth table for Contrapositive of the conditional statement “If p, then q” is given below:

Similarly, the truth table for the converse of the conditional statement “If p, then q” is given as:

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Converse Meaning in Geometry – Understanding Theorem Inverses

The converse in geometry refers to a form of statement that arises when the hypothesis and conclusion of a conditional statement are switched.

In a typical conditional statement of the form “If $p$ then $q$”, the converse would be “If $q$ then $p$”.

Understanding the converse is critical because it does not necessarily hold the same truth value as the original statement, which is a fundamental aspect of logical reasoning in geometry.

For instance, if we consider a simple conditional statement , such as “If a shape is a square, then it has four equal sides”, its converse would be “If a shape has four equal sides, then it is square”.

The converse may or may not be true, and assessing its validity is a key skill in geometric proofs and theorems. Let’s explore how the converse fits within the broader scope of mathematical reasoning and why it’s an intriguing concept that often challenges our initial assumptions.

Converse Meaning in Geometry

In geometry, when I work with a conditional statement , typically presented in the form “If p, then q,” there is a particular way to form a related statement known as the converse .

 Constructing the converse statement involves swapping the hypothesis (p) and the conclusion (q). The converse will thus read as “If q, then p,” formatted as $q \rightarrow p$.

Here’s a simple way to remember this:

I should be careful, though, because the truth of the converse is not guaranteed by the truth of the original conditional statement . It’s essential to examine it separately to see if it holds within the context of the geometry I am studying.

A special case arises when both a statement and its converse are true. In this scenario, the statement becomes a biconditional , symbolically represented as $p \leftrightarrow q$, pronounced as “p if and only if q.”

The exploration of logical equivalence in geometry leads me to two more related statements: the inverse and  the contrapositive .

The inverse flips both the hypothesis and the conclusion and applies negation to both, forming $ \neg p \rightarrow \neg q$. In contrast, the contrapositive statement negates and swaps the sides of the original, creating a statement that is always logically equivalent to the original, symbolized by $ \neg q \rightarrow \neg p$.

When examining theorems or propositions in geometry, understanding these related statements ensures a deeper comprehension of the concepts and their applications in logical reasoning.

Applications in Geometry

In geometry, converse , contrapositive statements , and the use of negations play crucial roles in understanding and proving theorems.

I often encounter the converse of a theorem, which involves reversing the hypothesis and conclusion of an if-then statement. For example, the converse of the statement “If a polygon is a square, then it has four right angles” would be “If a polygon has four right angles, then it is a square.”

Contrapositive statements , on the other hand, are formed by both negating and reversing the hypothesis and conclusion of the original conditional statement.

They are essential as they are logically equivalent to the original statement, which is a property I leverage frequently. Given the theorem “If a figure is a square, then it has four equal sides,” its contrapositive would be “If a figure does not have four equal sides, then it is not a square,” and both statements are true if one is true.

When examining the properties of geometric figures, these logic structures guide me to derive new theorems or validate existing ones.

Below is a table that illustrates the relationships between conditional statements and their converse and contrapositive forms using polygons:

As I progress through geometric problems, I pay special attention to these logical forms to make sure my conclusions are valid.

The meaning behind each statement often informs my approach to proving or disproving a given theorem . This systematic application of logic helps me and fellow mathematicians to rigorously establish truths within the realm of geometry.

In my exploration of converses within the realm of geometry , I’ve found that understanding these concepts enhances my logical reasoning and analytical skills.

A conditional statement , which is typically of the form “If $p$, then $q$,” where $p$ and $q$ are specific statements, lays the foundation for these discussions. The converse flips this relationship to “If $q$, then $p$.”

What captivates me is that the truth of a converse is not guaranteed by the original conditional statement .

For example, if the statement is “If a figure is a square, then it has four sides,” the converse would be “If a figure has four sides, then it is a square,” which isn’t necessarily true, as the figure could also be a rectangle or any other quadrilateral .

The utility of knowing that a conditional and its converse can create a biconditional statement —when both are true—is apparent in proofs, where we might state that “$p$ if and only if $q$” or $p \leftrightarrow q$.

Remembering that the converse of a conditional statement is just one part of the puzzle, with the inverse and contrapositive also adding depth to the study of statements, settles my curiosity for the moment.

These relationships between statements are invaluable tools in the logical structure that underpins much of geometry .

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Converse Statement: Definition and Explanation

The three most common ways to change a conditional statement are by taking its inverse, its converse statement, or its contrapositive. In each case, either the hypothesis or the conclusion swap places, or a statement is replaced by its negation.

Table of Contents

The Inverse Statement

A conditional statement is reversed by replacing the hypothesis and the conclusion with their negations. In the case of the statement, “A vertex that is confined to a circle is called an inscribed angle”, the converse of the statement is “A vertex that isn’t confined to a circle is called an ungrounded angle.” Both the hypothesis and the conclusion were negated. If the original statement reads “if  j , then  k “, then the reverse would be, “if not j, then not k.”

It is not possible to determine the true inverse of a statement. The inverse of some statements may have the same truth value as the inverse of another statement and vice versa. The statement “A four-sided polygon is a quadrilateral” and its inverse, “A polygon with more or fewer sides than four sides is not a quadrilateral,” are both true (the truth value of each is T). The original statement and its inverse, however, are not equally true in the inscribed angles example above. It is true that an angle with its vertex on a circle will not be an inscribed angle, but the converse is false: it is possible for an angle with its vertex on a circle to still not be an inscribed angle.

The Converse Statement

By switching the hypothesis and conclusion, the converse of a statement can be formed. Conversely, “If two lines don’t intersect, then they are parallel” is “If two lines are parallel, then they don’t intersect.” The converse of “if p, then q” is “if q, then p.”

A statement’s converse may or may not have the same meaning as its original. A tiger is a mammal, for instance, so the converse would be a mammal is a tiger. That’s certainly not the case.

In all cases, however, a definition must have a contrarian interpretation. If this is not the case, then the definition is not valid. For example, we know the definition of an equilateral triangle well: “if all three sides of a triangle are equal, then the triangle is equivalent.” As a result, “If a triangle is equivalent, it has three equal sides.” But what if we put this test to the test with a wrong definition on it? If we incorrectly stated the definition of a tangent line as: “A tangent line crosses a circle,” the statement would be true. However, the converse, “A line that intersects a circle is a tangent line”, is incorrect; the converse could describe a secant line as well. Therefore, the converse is very helpful in determining the validity of a definition.

What Is A Converse Statement?

A converse statement is obtained by reversing the hypothesis and conclusion of a conditional statement.

Explanation

Understanding “hypothesis” and “conclusion” is important.

A statement which is of the form of “if p then q” is a conditional statement, where ‘p’ is called the hypothesis and ‘q’ is called the conclusion.

If something is negated, it means it conveys the opposite of that thing.

Converse is a statement derived by reversing a statement’s hypothesis and conclusion.

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Everyday life and its variability influenced human evolution at least as much as rare activities like big-game  hunting

Professor and Chair of Biology at Seattle Pacific University and Affiliate Assistant Professor of Anthropology, University of Washington

Disclosure statement

Cara Wall-Scheffler does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.

University of Washington provides funding as a member of The Conversation US.

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Think about taking a walk: where you need to go, how fast you need to move to get there, and whether you need to bring something along to carry the results of your errand.

Are you going on this walk with someone else? Does walking with a friend change your preparation? If you’re walking with a child, do you remember to bring an extra sweater or a snack? You probably did – because people intuitively vary their plan depending on their current needs and situations.

In my research as an anthropologist , I’ve focused on the evolution of human walking and running because I love the flexibility people bring to these behaviors. Humans in all kinds of environments across space and time vary how far they go, when they go and what they go for – whether food, water or friends – based on a multitude of factors, including season, daylight, rituals and family.

Anthropologists split their studies of human activity into two broad categories: what people need to do – including eat, keep their kids alive and so on – and what solutions they come up with to accomplish these needs.

How people keep their children alive is a key issue in my research because it has a direct impact on whether a population survives. It turns out that kids stay alive if they’re with adults. To this end, it is a human universal that women carry heavy loads every day , including kids and their food. This needs-based behavior seems to have been an important part of our evolutionary history and explains quite a few aspects of human physiology and female morphology , such as women’s lower center of mass .

The solutions to other key problems, like specifically which food women will be carrying, vary across time and space. I suggest that these variations are as integral to explaining human biology and culture as the needs themselves.

Impacts of uncommon activities

Evolutionary scientists often focus on how beneficial heritable traits get passed on to offspring when they provide a survival advantage. Eventually a trait can become more common in a population when it provides a useful solution.

For example, researchers have made big claims about how influential persistence hunting via endurance running has been on the way the human body evolved. This theory suggests that taking down prey by running them to exhaustion has led to humans’ own abilities to run long distances – by increasing humans’ ability to sweat, strengthening our head support and making sure our lower limbs are light and elastic.

But persistence hunting occurs in fewer than 2% of the recorded instances of hunting in one major ethnographic database , making it an extremely rare solution to the need to find food. Could such a rare and unusual form of locomotion have had a strong enough impact to select for the suite of adaptive traits that make humans such excellent endurance athletes today?

Maybe persistence hunting is actually a fallback strategy, providing a solution only at key moments when survivorship is on the edge. Or maybe these capabilities are just side effects of the loaded walking done every day . I think a better argument is that the ability to predict how to move between common and uncommon strategies has been the driver of human endurance capacity.

Everyday life’s influence on evolution

Hunting itself, especially of large mammals, is hardly ubiquitous , despite how frequently it is discussed. For example, anthropologists tend to generalize that people who lived in the Arctic even up to a hundred years ago consumed only animal meat hunted by men. But actually, the original ethnographic work reveals a far more nuanced picture .

Women and children were actively involved in hunting, and it was a strongly seasonal activity. Coastal fishing, berry picking and the use of plant materials were all vital to Arctic people’s day-to-day sustenance. Small family groups used canoes for coastal foraging for part of the year.

During other seasons, the whole community participated in hunting large mammals by herding them into dangerous situations where they were more easily killed. Sometimes family groups were together, and sometimes large communities were together. Sometimes women hunted with rifles, and sometimes children ran after caribou.

The dynamic nature of daily life means that the relatively uncommon activity of hunting large terrestrial vertebrates is unlikely to be the main behavior that helps humans solve the key problems of food, water and keeping children alive.

Anthropologist Rebecca Bliege Bird has investigated how predictable food is throughout the day and the year . She’s noted that for most communities, big game is rarely caught, especially when a person is hunting alone. Even among the Hadza in Tanzania, generally considered a big-game hunting community, a hunter acquires 0.03 prey per day on average – essentially 11 animals a year for that person.

Bird and others clearly argue that the planning and flexible coordination done by females is the crucial aspect of how humans survive on a daily basis. It’s the daily efforts of females that allow people to be spontaneous a few times a year to accomplish high-risk activities such as hunting – persistence or otherwise. Therefore it is female flexibility that allows communities to survive between the rare big-game opportunities.

Changing roles and contributions

Some anthropologists argue that in some parts of the world, behavior varies more for cultural reasons , like what tools you make, than for environmental ones, such as how much daylight there is during winter. The importance of culture means that the solutions vary more than the needs.

One of the aspects of culture that varies is the role assigned to specific genders. Varying gender roles are related to the distribution of labor and when people take on certain solution-based tasks . In most cultures, these roles change across a female’s life span. In American culture, this would be like a grandparent going back to college to hone a childhood passion in order to take on a new job to send their grandchildren to college.

In many places, females go from youth when they might carry their siblings and firewood, to early parenthood where they might go hunting with a baby on their back , to older parenthood where they might carry water on their head, a baby on their back and tools in their hands, to postmenopausal periods when they might carry giant loads of mangoes and firewood to and from camp.

Even though always load carrying , our capacity to plan and change our behavior for diverse environments is part of what drives Homo sapiens ’ success, which means that the behavior of females across their different life stages has been a major driver of this capability.

• Anthropology
• Human evolution
• Ethnography
• Women's bodies

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