Let $f:A \rightarrow B$ be a function.

Define a relation $\sim$ on $A$ by $a\sim b$ iff $f(a)=f(b)$.

a) Show that $\sim$ is an equivalence relation on $A$.

b) If $A_{\sim}$ is the set of equivalence classes $\{[a]|a \in A\}$, show that the function $h:A_{\sim} \rightarrow B$ defined by $h([a])=f(a)$ is $1-1$.

I had no problem solving part a) but am confused on what to do for part b). Thanks

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    $\begingroup$ As an extra, show that $A=A_{\sim}$ if and ony if $f$ is injective. $\endgroup$ – Mathematician 42 Jan 21 '17 at 22:06

Suppose $h([a])=h([b])$, we have $f(a)=f(b)$.

From definition of the relation$\sim$, we have hence we have $a \sim b$ and hence $[a]=[b]$.

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  • $\begingroup$ thank you so much. I was really close, just didn't think it was that simple! $\endgroup$ – Nicole Jan 22 '17 at 3:59

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