I am trying to show that equality of cardinality is an equivalence relation.
I am a bit confused as to how I should approach this.
Currently, my understanding of equivalence relation is that a relation needs to be
reflexive, symmetric, and transitive in order to be an equivalence relation.
For instance, if I have
$A = \{1,2,3,5\}$
$R = \{(1,1),(2,2),(3,3),(5,5),(1,5),(5,1)\}$, $R \subseteq A \times A$
$R$ would be an equivalence relation.
If relation $R$ is an equality of cardinality, I am not sure as to how I can
show that it is reflexive, symmetric, and transitive.
Could somebody help me?
Hint. By definition, two sets $A$ and $B$ have the same cardinality, hence $A\sim B$, if there exists a bijection $A\to B$. On the other hand if we have a bijection $A\to B$ and $B\to C$, we need to show there is a bijection $A\to C$, which would mean that $A\sim B$ and $B\sim C$ implies $A\sim C$ (transitivity); can you try to do so using the fact that composition of bijections is a bijection? To show it is symmetric, given a bijection $A\to B$ we need to find a bijection $B\to A$, which should be easy to show, noting the fact that bijections have inverses which are also bijections. Finally to see $A\sim A$ (reflexivity) we just need to find a bijection from $A$ to itself, which should be easy to see.