Questions on equivalence relation and functions

Question

I just found this question in my discrete math homework and just can't have the solution by looking through the textbook.

The question contains two parts:

a) If $R$ is an equivalence relation on set $A$, is $R$ necessarily a function $A\rightarrow A$?

b) If $R$ is an partial order relation on set $A$, is $R$ necessarily a function $A\rightarrow A$?

Can anyone give some tips? I hope you can tell from the basics.

Answer 1

Let $A = \mathbb N$.

For a) if $R = \{(a,b) \mid a = b \mod 2\}$, can you find two different values $x,y \in \mathbb N$ such that $(2,x) \in R$ and $(2,y) \in R$?

Can you find a similar example for b)?