Let S, T and P be three nonempty set. Prove that (a)S~S (b)If S~T, then T~S

Question

(a) S~S means it is reflexive (b) If S~T, then T~S means it is symmetry Using the definition of equivalent sets, set S is equivalent to T if and only if there exists a function f:S->T which is one-to-one and onto. set T is equivalent to T if and only if there exists a function f:T->S which is one-to-one and onto. How to prove (a)S~S (b)If S~T, then T~S

Answer 1

I'll use $S eq T$ to mean $S$ is equivalent to $T.$ Ok, so then for part (a) of your problem, $S eq S$ means there is a function $f:S \to S$ which is one-to-one and onto. To finish that claim you only need one such bijection $f,$ and the first thing one would try is the identity mapping $f:S \to S$ defined by $f(s)=s$ for each $s.$