Prelim Problem 21 J.O

21. (Preliminary for T, March 13; finalized Sat, March 17) Given a partition {Xα}α∈I of X, show that the relation R⊆X×X defined by ((a,b)∈R⇔∃α∈I such that both a,b∈Xα) is an equivalence relation on X whose equivalence classes are precisely the sets Xα of the partition.

Pf:

To prove reflexivity: Since $[(a,b)]=[(a,b)], (a,b)\sim (a,b)$ and since $\{X_{\alpha}\}$ is a partition and $\bigcup\{X_{\alpha}\}_{\alpha \in I} = X$, this is true.
To prove symmetry: suppose there is some $(c,d)\in R$ which implies $\exists \alpha \in I$ such that both $c,d\in X_{\alpha}$.
If $[(a,b)]=[(c,d)]$, then $[(c,d)]=[(a,b)]$ which implies that $(a,b)\sim (c,d)$ implies $(c,d)\sim (a,b)$.
To prove transitivity: Suppose there is another element $(e,f)\in R$ which implies $\exists \alpha \in I$ such that both $e,f\in X_{\alpha}$.
If $[(a,b)]=[(c,d)]$ and$[(c,d)]=[(e,f)]$, then it is implied that if $(a,b)\sim (c,d)$ and $(c,d)\sim (e,f)$ then $(a,b)\sim (e,f)$ is implied.

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