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
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.
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.
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.
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!
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