Transitive Property: Definition & Examples

Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University, and her Bachelor's degree in Mathematics from Grand Valley State University. She has 20 years of experience teaching collegiate mathematics at various institutions.

The transitive property in mathematics states that if a = b and b = c, then a = c. Explore the definition and examples of the transitive property and learn about cautions for the transitive property. Updated: 10/15/2021

The Transitive Property

Let's look at a question. If you were told that Maria has the same mom as Doug, and Doug has the same mom as Sara, is it safe to say that Maria has the same mom as Sara? You might be thinking that of course it's safe to say this - it's simple logic, but it doesn't hurt to stop and make sure you're not missing something. Don't worry! You're not! This isn't a trick question, and you are right in saying that it is safe to say that Maria has the same mom as Sara.

The logic behind this assessment has to do with the transitive property in mathematics. The transitive property states that:

If a = b and b = c, then a = c

Another way to look at the transitive property is to say that if a is related to b by some rule, and b is related to c by that same rule, then it must be the case that a is related to c by that rule.

In looking at the transitive property in this way, we see why it makes perfect sense that if Maria has the same mom as Doug, and Doug has the same mom as Sara, then it must be the case that Maria has the same mom as Sara. Let's take a closer look at the transitive property and its uses in both mathematics and real-world instances.

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  • 0:03 The Transitive Property
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Cautions for the Transitive Property

Although the transitive property seems pretty straight forward, there are some things to be careful of when using it to avoid errors in logic and misuse of the property. For instance, suppose three people are running in a race. Call them persons A, B, and C. In the race, they finish in the order A, B, C.

Now, suppose a friend of yours missed the race but heard that person A beat person B, and person B beat person C. They ask you if person A beat person C. You tell them that they already have all the information they need to answer their own question.

By the transitive property, in this specific race, A beat B, and B beat C, so it must be the case that A beat C. This is a correct use of the transitive property, and all the logic involved is correct. However, suppose your friend says that what they meant to ask was if person A would beat person C in an upcoming race. If you were to reply that person A would beat person C because the transitive property says that A beat B, and B beat C, this would not be the correct use of the transitive property.

The reason why this is incorrect is because this race is an isolated incident. The results don't guarantee that A will always beat B or B will always beat C, so we can't guarantee that A will always beat C. In other words, we can't apply the transitive property to answer your friend's question about an upcoming race. The property can only be applied to that specific incident. Therefore, we see that though the transitive property is fairly simple and straightforward, we have to be careful when, where, and how we apply it.

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