1. ## Equivalence Relation ~

I'm having trouble with these types of problems. Any help would be greatly appreciated .

Let $\displaystyle A_1 = \{0,1\}, A_2 = \{1,2\}, A_3 = \{3,4\}$ be subsets of $\displaystyle S = \{0,1,2,3,4\}$. Explain why the relation ~ on S defined by x~y if and only if $\displaystyle x,y \in A_i$ for some $\displaystyle i \in \{1,2,3\}$ is NOT an equivalence relation on S.
I know that in order ~ to be an equivalence relation, ~ must be:
1) (reflexive) x~ x for all x in X.
2) (symmetric) If x~ y then y~ x.
3) (transitive) If x~ y and y~ z, then x~ z.
But, I just can't find a counter example for which it doesn't satisfy one or more of 1,2,3.

2. ## Equivalence Relation

Hello shinn

Let be subsets of . Explain why the relation ~ on S defined by x~y if and only if for some is NOT an equivalence relation on S.

I know that in order ~ to be an equivalence relation, ~ must be:
1) (reflexive) x~ x for all x in X.
2) (symmetric) If x~ y then y~ x.
3) (transitive) If x~ y and y~ z, then x~ z.
But, I just can't find a counter example for which it doesn't satisfy one or more of 1,2,3.
(1) and (2) are both satisfied. So what about (3)? $\displaystyle 0\sim 1$ because $\displaystyle 0, 1 \in A_1$ and $\displaystyle 1\sim 2$ because $\displaystyle 1, 2 \in A_2$. But $\displaystyle 0\sim 2$?