..An equivalence relation on $S$ is determined by the subset $R$ of $ S \times S$ consisting of those pairs $\left(a,b\right)$ such that $a \sim b$. Write the axioms for an equivalence relation in terms of the subset $R$.
Since I was not getting any clue I took $S$ to be $\{1,2,3,4\}$ and defined $a\sim b$ if $a+b=2k$. Clearly this is an equivalence relation. Then $S=\{(1,1),(1,3),(3,1),(2,2),(2,4),(4,2),(4,4)\}$.
Then I defined $ a\sim b$ if $a \lt b$. This relation is only transitive. Here $S=\{(2,1),(3,1),(3,2),(4,1),(4,2),(4,3)\}$.
Again I defined $ a\sim b$ if $a \le b$. This relation is reflexive and transitive but not symmetric. Here $S=\{(2,1),(3,1),(3,2),(4,1),(4,2),(4,3),(1,1),(2,2),(3,3),(4,4)\}$
Based on these I defined the following: $\sim$ is said to be an equivalence on $S$ if the following hold:
(i) $(a,a) \in R$ for all $a$ in $S$ (Reflexive)
(ii) If $(a,b) \in R$, then $(b,a) \in R$ for all $a$ and $b$ in $S$(Symmetric)
(iii) If $(a,b) \in R$ and $(b,c) \in R$, then $(a,c) \in R$ for all $a$, $b$ and $c$ in $S$ (Transitive)
Is this the correct one??
Thanks for the help!!
A question from Michael Artin - Algebra (2nd Edition), Exercise 2.7.2
The equivalence relation in the question is abstract and needs no specifying, it’s purely a practice for realising the three axioms (transitive, symmetric and reflexive) in terms of the subset $R$. $R$’s 'being the subset $R$ of $ S \times S$ consisting of those pairs $\left(a,b\right)$ such that $a \sim b$' stipulates the form for realisation. I.e. we know that $a \sim b$ is written as $\left(a,b\right)$, hence other expressions follow.
Your answer is correct but the exemplification is IMO unnecessary, and I should have commented rather than written an answer had it not been for the lack of reputations