Thread: discrete math - equivalence relation

1. discrete math - equivalence relation

Let A be a set. For every set B and total function f:A->B we define a relation R on A by R={(x,y) belonging to A*A:f(x)=f(y)}

*belonging to - because i dont know how to make the symbole....

Prove that f is one-to-one if and only if the equivalence classes of R are all singletones

2. There is a class with more than one element if, and only if there are $x,y \in A,\ x\neq y$ such that $f(x)=f(y)$, and that means $f$ is not one-to-one.

Since for any propositions $P,Q$, $(P\Leftrightarrow Q )\Leftrightarrow (\lnot P\Leftrightarrow \lnot Q)$, we've finished.

(just two contrapositions)