## 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.

## Examples

Let's look at some additional examples.

Suppose you're in math class, and the teacher tells you that a line *L1* is parallel to another line *L2*. She then goes on to say that line *L2* is parallel to a third line *L3*. As she continues lecturing, you stop to think. What can you say about the relationship between line *L1* and line *L3*?

Ah-ha! This is a case for the transitive property! Since *L1* is parallel to *L2* and *L2* is parallel to *L3*, it must be the case that *L1* is parallel to *L3*. You raise your hand to tell the teacher, and she's quite impressed with your observation!

Okay, one more time!

Suppose you were told that teams 1, 2, and 3 are playing in a tournament. In the first game, team 1 beat team 2. In the second game, team 2 beat team 3. In the third game, team 1 will play team 3. Since team 1 beat team 2, and team 2 beat team 3 in the first two games, can we use the transitive property to conclude that team 1 will beat team 3 in the third game? If you're thinking no, then you're right. This is one of those instances where we can't apply the transitive property! How team 1 will fare against team 3 can't be guaranteed based on the previous 2 games (otherwise, gambling and betting would be a lot easier, as would fortune telling!). This is another example illustrating that we have to be careful not to use the transitive property inappropriately.

## Lesson Summary

In mathematics, the **transitive property** states that:

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

In other words, if *a* is related to *b* by some property, and *b* is related to *c* by the same property, then *a* is related to *c* by that property. The transitive property is a straightforward and simple property, and it can be used in many real-world applications. However, it pays to be careful when determining whether or not it applies to some instances. The more we work with it, the easier this determination will become, so make sure to keep practicing!