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I'm looking at a True or False question in my book and it is very close to identical to the definition of the transitive property in the book, though this answer is False. If someone could explain to me why this is False, I would be very appreciative.

Here's the True or False Statement:

Let R be a binary relation on a set A. If there exist a,b,c within A such that (a,b) within R, (b,c) within R, and (a,c) within R, then R must be transitive.

Here's the definition from the book:

A binary relation R on a set A is transitive if and only if

if a,b,c is within A, and both (a,b) and (b,c) are in R, then (a,c) is within R

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The mere existence of elements three elements $a, b, c$ such that $(a, b) \in R, (b, c) \in R, \text{ and } (a, c)\in R$ does not guarantee that transitivity holds for all pairs of elements

For example: Let's say our set is $\{a, b, c,\}$, and let $$R = \{(a, b), (b, c), (a, c) , (b, a)\}.$$

We have that $(b, a)\in R$ and $(a, b) \in R$, but $(b, b)\notin R$. We also have $(a, b) \in R, (b, a) \in R$, but $(a, a) \notin R$. So transitivity fails, even though we do have $(a, b) \in R, (b, c) \in R$, and $(a, c) \in R$. Another way to put it is that knowing $(a, b), (b, c), (a, c)\in R$ is not sufficient to conclude that the relation is transitive.

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In the statement, you are only given that there is some $a,b,c$ that satisfies the requirement. From the definition of transitive, you need it to be true for any $a,b,c$. So let $A=\{1,2,3,4\}, R=\{(1,2),(2,3),(1,3),(2,4)\}$ We are missing $(1,4)$ in $R$ for transitivity, but the first three elements listed in $R$ meet the True/False statement.

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