# Discrete math - equivalence relation

## Main Question or Discussion Point

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

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You can write TeX code by inserting [ tex] ... [\ tex] and putting the code in between (obviously without the spaces). The notation for element of is \in: $$x \in A$$.

You can hit quote to see the code behind what I wrote.

Have you made any attempt at this problem? It should follow easily from the definitions.

MathematicalPhysicist
Gold Member
OK, here is one attempt.
f is 1-1 iff for every x,y if f(x)=f(y) then x=y.

The relation is defined by ~<=> x~y<=>f(x)=f(y), thus the equivalence class [x]={y|y~x}={y|f(x)=f(y)} if it's one to one it's evident that [x] is a singleton.

MathematicalPhysicist
Gold Member
Btw, it's not in the right place this thread of yours, it should be in the set theory section.