How it can be shown that:
The equality equivalence relation is the finest equivalence relation relation.
Since the equality relation denoted $R$ is an equivalence relation hence it should be a subset from the Cartesian product of two same sets namely $X$,e.g.:
$$R⊆X×X$$
I know that we should pick an ordered pair $\left(x,y\right)∈R$ where $x,y∈X$ and show if it's an element of the equality equivalence relation $R$ then it's also contained in any other equivalence relations and using the definition of subset the result follows, but I don't know how exactly it can be done.
The other theorem states:
The Trivial relation that makes all pairs of elements related is the coarsest relation.
The trivial relation denoted $R$ is a subset of Cartesian product of two subsets $S,T$, e.g. $$R⊆S×T$$ and since every element in $R$ is related to the other elements so $R=S×T$
I don't know how to proof this one and any help is appreciated. (Trivial relation is not necessarily a subset of Cartesian product of two same sets, hence it's not necessarily a homogeneous binary relation)