Show that a set $A/{\sim}$ has the same cardinality as $B$ Let $A$, $B$ be sets. $f\,\colon A\rightarrow B$ is a surjective function. Define a relation $\sim$ in A by:
for every $x$, $y\in A$, $x\sim y$ if $f(x)=f(y)$
(a) Show that $\sim$ is an equivalence relation.
(b) Show that $A/{\sim}$ has the same cardinality as $B$.
I have finished (a) but do not know how to start with part (b). What does $A/{\sim}$ mean? Does it mean that exclude the elements satisfying $\sim$ and in set $A$?
 A: Hint: Let $\pi\colon A\to A/{\sim}$ be the canonical projection. Show that $\tilde{f}\colon A/{\sim}\to B$ defined by the relation $\tilde{f}\circ\pi=f$ is bijective.
A: I haven't posted an answer here because Dietrich Burde already posted the answer in a comment, but some hours have passed and he hasn't made it into an answer.
$A/{\sim}$ is the set of equivalence classes.
For eample, suppose $A=\{a,b,c,d,e\}$ and $a,b,c$ are equivalent to each other, and $d,e$ are equivalent to each other but not to $a$ or $b$ or $c$.  Thus
$$
a\sim b\sim c \quad \text{and}\quad d\sim e.
$$
Then $A/{\sim}$ has two members:
$$
A/{\sim} = \big\{\  \{a,b,c\},\  \{d,e\}\  \big\}.
$$
It is as if we are regarding $a,b,c$ as being just one thing rather than three, and that one thing is one member of $A/{\sim}$ and there is one other member, called either $d$ or $e$.
Or one could say that $a,b,c$ are different in their roles as members of $A$, but are the same as each other in their roles as members of $A/{\sim}$.
