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Definition (Equivalence Relation)
Let $S$ be set. Let $\cong$ be a relation on $S$, meaning $\cong$ is a collection $\mathscr{C}$ of ordered pairs of $S$. We way that $A\cong B$ if and only if the ordered pair $(A,B)$ is an element of $\mathscr{C}$. We say that $\cong$ is an equivalence relation if and only if
- (Reflexive) For every $A\in S$, we know that $A\cong A$ (so the ordered pair $(A,A)$ is always in $\mathscr{C}$).
- (Symmetric) If $A\cong B$, then $B\cong A.$
- (Transitive) If $A\cong B$ and $B\cong C$, then $A\cong C$.
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