# Transitive Property of Equality

# Transitive Property of Equality

**Transitive Property: **The *transitive property *meme comes from the *transitive property* of equality in mathematics. In math, if A=B and B=C, then A=C. So, if A=5 for example, then B and C must both also be 5 by the *transitive property*. This is true in—a foundational property of—math because numbers are constant and both sides of the equals sign must be equal, by definition. (Hey, we’re word people, but we know a thing or two about definitions.)

The *transitive property *slang/meme, on the other hand, misapplies the *transitive property* to non-numerical things to reach illogical conclusions or false equivalencies. For example, humans eat cows and cows eat grass, so by the *transitive* property, humans eat grass. Unlike in math, just because the first two statements are true does not make the final “conclusion” true. The humor in the meme relies on the absurdity of attempting to use the *transitive property* outside of math.

## Transitive Property Of Congruence

The meme apparently evolved out of the practice of sports fans attempting to apply the *transitive property *to sports teams and athletes. While it almost certainly predates the internet, as far back as 1992 sports fans were attempting to argue their team was the best using the *transitive property* on Usenet groups. It has reached the point of a sports cliché, with web sites like myteamisbetterthanyourteam.com created just to mock it.

Through the 1990s, internet users started using *transitive property *to “prove” the superiority of other things, such as the best pilot in *Star Wars* or the most powerful character in *Lord of the Rings*. By the the 2000s, the *transitive property *was used more generally in silly internet arguments (or intentional punchlines) that tried to connect two seemingly unrelated things.

## Transitive Property Definition

In this video lesson, we talk about the **transitive property of equality**. This property tells us that if we have two things that are equal to each other and the second thing is equal to a third thing, then the first thing is also equal to the third thing.

You can think about it in terms of identical toy trucks. Say you have two identical toy trucks. They are both blue. Now, if the second truck is the same as a third blue toy truck, then we can also say that the first truck is also the same as the third blue toy truck. This is because we know that the first toy truck is equal to the second toy truck, so if either of those toy trucks is equal to a third, then they both are equal to that third truck. It’s like a chain. They are all linked to each other.

In math, we have a formula for this property. It says that if *a* = *b* and *b* = *c*, then *a* = *c*. This is telling us that if two things are equal and the second thing is equal to a third, then because the first two things are equal, it also means that the first is equal to the third as well. They are all equal to each other. Let’s look at a couple of examples to see how this transitive property of equality works in action.

### Example 1

In this example, we look at how true the transitive property of equality is. We begin with our two equations:

5 = 3 + 2 and 3 + 2 = 5

We can label these two equations with our letters. Or, we can use our toy trucks. The first 5 is our letter *a*, or our first toy truck. The following 3 + 2 is our letter *b* or our second toy truck. The last 5 is our letter *c*, or our third toy truck. By applying the transitive property of equality, we can say that the first 5 is equal to the last 5, that letter *a* is equal to letter *c*, or that the first toy truck is equal to the third toy truck. Is this true? Let’s see.

5 = 5

The following property: If *a* = *b* and *b* = *c*, then *a* = *c*. One of the equivalence properties of equality.

Note: This is a property of *equality* and *inequalities*. (Click here for the full version of the transitive property of inequalities.) One must be cautious, however, when attempting to develop arguments using the transitive property in other settings.

Here is an example of an unsound application of the transitive property: “Team A defeated team B, and Team B defeated team C. Therefore, team A will defeat team C.”

## What Is 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 **in mathematics. The transitive property states that:

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

Another way to look at the transitive 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.

### Cautions for the Transitive Property

Although the 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 that 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 property is fairly simple and straightforward, we have to be careful when, where, and how we apply it.

## What is an example of the transitive property?

The **transitive property** meme comes from the **transitive property** of equality in mathematics. In math, if A=B and B=C, then A=C. So, if A=5 for **example**, then B and C must both also be 5 by the **transitive property**. For **example**, humans eat cows and cows eat grass, so by the **transitive property**, humans eat grass.

## What is the transitive property in geometry?

**Transitive Property** (for three segments or angles): If two segments (or angles) are each congruent to a third segment (or angle), then they’re congruent to each other. The **Transitive Property** for three things is illustrated in the above figure.

## How do you explain transitive property?

**transitive property**of equality tells us that if we have two things that are equal to each other and the second thing is equal to a third thing, then the first thing is also equal to the third thing. The formula for this

**property**is if a = b and b = c, then a = c.