Conceptual question about equivalent relations and bijections? 
*

*What is the correlation between equivalence relation and bijection? 

*Is equivalence relation a synonym for bijection? 

*Does equivalence relation always imply there exist a bijection?  



I was reading Definition 2.3 and 2.4. I got out of it that equivalent relations are somehow related to bijections.
 A: I would say that the excerpt you've cited demonstrates poor pedagogy by introducing a concept as important as equivalence relations in such a comment.
What the excerpt is saying is that the "relation" on the "set of all sets" (warning: there is no such thing, that is why I am using scare quotes) defined by
$$A\sim B\iff\text{ there exists a bijection }f:A\to B$$
is an equivalence relation. While it is important to observe this property, it is by no means the only thing there is to say about bijections, nor is it the only example of an equivalence relation.

Equivalence relations are not related to bijections. 
Given a set $X$, a relation on $X$ is a subset $R$ of the set $X^2=\{(x,y):x,y\in X\}$. We say that $R$ is an equivalence relation if it has these properties:


*

*$(x,x)\in R$ for all $x\in X$.

*$(x,y)\in R$ $\implies$ $(y,x)\in R$

*$(x,y)\in R$ and $(y,z)\in R$ $\implies$ $(x,z)\in R$


Given sets $X$ and $Y$, a bijection from $X$ to $Y$ is a function $f:X\to Y$ with the property that for all $y\in Y$, there exists exactly one $x\in X$ such that $f(x)=y$. Equivalently, $f:X\to Y$ is a bijection if there exists a function $g:Y\to X$ such that $(g\circ f)=\mathrm{id}_X$ and $(f\circ g)=\mathrm{id}_Y$.
(If one interprets a function $f:X\to Y$ as a subset of $X\times Y$, then technically one can compare equivalence relations on a set $X$ with functions $X\to X$; the only function that is an equivalence relation, and the only equivalence relation that is a function, is the identity function $\mathrm{id}_X:X\to X$.)
It sounds like what you're thinking of are partitions and equivalence relations, which are equivalent mathematical notions, as described by the "fundamental theorem of equivalence relations" (Wikipedia link).
A: What these definitions say is that one particular equivalence relation on the class of sets can be defined by the means of bijections: in a precise way, sets $A$ and $B$ are equivalent is there exists a bijection from $A$to $B$. 
But a general equivalence relation is a relation that satisfies the three properties that @Zev Cholones recalled. And indeed the equivalence between sets defined with bijections (it is also called equipotence) satisfies them all:


*

*$A\sim A$, because $\operatorname{id}_A$ is trivially a bijection.

*If $A\sim B$, i.e. if there exists a bijection $f\colon  A\to B$, then $B\sim A$, i.e. there exists a bijection $g\colon B\to A$: it is enough to take the reciprocal of $f$, which is also a bijection.

*If $A\sim B$ and $B\sim C$, then $A\sim C$: Indeed if $f\colon A\to B$  and $g\colon B\to C$  are bijections, then $g\circ f\colon A\to C$ is a bijection.

