Here is a question where I have to prove if this relation is a equivalence relation or not? I also have to provide necessary proof for the same. I present to you my question and my solution.
% is a relation on the set of all subsets of N, A%B iff $$A\div B $$ is a finite set. We need to determine if % is an equivalence relation or not?
Reflexive relation:
$$A R A$$ $$as\ (A \div A) = (A \bigcup A) -(A \bigcap A) = \{\} $$ and I considered null set as a finite set, I know this could be a mathematical disaster, but my instructor gave me this hint and I hope i haven't misinterpreted him.
Symmetric relation:
$$A R B \implies BRA$$ $$A \div B \implies B \div A$$ $$ \{ \Bbb N \} \implies \{\Bbb N\} $$ And similar for transitive relation. I was a bit confused with his hint, please correct me for any wrong doing. Thanks!
I'll show you how to prove symmetry, and hopefully that will show you how to approach transtivity yourself. Suppose we have $A$%$B$. Then that means that $A\cup B$ - $A \cap B $ is a finite set $D$. But since $B \cup A = A \cup B$ and $B \cap A = A \cap B$, we have that $B \cup A - A \cap B=D$, and is therefore finite as well. Hence, whenever $A$%$B$, we have $B$%$A$, and hence $R$ is symmetric.
HINT for transitivity: draw a Venn diagram!