I am asked: Determine whether the relation X on the set Z is reflexive, symmetric, antisymmetric, and/or transitive, where $(a,b) ∈ X$ if and only if a = 1.
Apparently, the set is antisymmetric and transitive. I understand how it is not reflexive or symmetric, but I don't get why it is transitive. If the set is composed of ${(1,0),(1,1),(1,2),(1,3),...}$, where is the transitivity found?
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You want to show that if $a \sim b$ and $b \sim c$ then $a \sim c$, where I've written $x \sim y$ for the statement that $(x,y) \in X$.
If $a \sim b$, then $a=1$. Hence $a \sim c$ for all $c$. We didn't actually need to assume that $b \sim c$.
For completeness, I'll go over why it's antisymmetric and not reflexive.
It's antisymmetric: if $a \sim b$ then $a=1$, while the only way $b \sim a$ is if $b=1$. Hence if both $a \sim b$ and $b \sim a$, then $a=b=1$.
It's not reflexive: $1 \sim 2$ but $2 \not \sim 1$.
Let's say we have $(a, b) \in X$ and $(b, c) \in X$. Since $(a, b) \in X$, we have $a=1$. Since $a=1$, we have $(a, c) \in X$, proving transitivity.
The interesting thing here is that because we have $(b, c) \in X$, we have $b=1$. Therefore, transitivity only applies when $(a, b)=(1, 1)$.