Equivalence Relations and functions on partitions of Sets

Question

let $f$ be a function on $A$ onto $B$. Define an equivalence relation $E$ in $A$ by: $aEb$ if and only if $f(a)=f(b)$.

Define a function $\phi$ on $A/E$ by $\phi([a]_{E})=f(a)$.

Hint: Verify that $\phi([a]_{E})=\phi([a']_{E}) $ if $[a]_{E} = [a']_{E}$

I know that A/E is a partition of A.

I know that a binary relation $F$ is called a function if $aFb$ and $aFc$ implies $b=c$ for any $a, b, \text { and } c$

Answer 1

To show that $\phi$ is a function, you must follow the hint. If $[a]_E=[a']_E$, then what can we say about $f(a)$ and $f(a')$? What then can be said about $\phi\left([a]_E\right)$ and $\phi\left([a']_E\right)$? You'll need to keep the definitions of $E,\phi$ in mind.

Answer 2

Hint: Suppose that $\langle [a]_E,x\rangle\in \phi$ and $[a]_E=[a']_E$, then $x=f(a)=f(a')$, conclude the wanted conclusion.

Bonus points: Show that $\phi$ is injective!

More bonus points: Show that every equivalence relation is actually induced like this by some function $f$.

(If you do both the bonus points thingies (they are quite short actually), then you have completely understood the topic.)