Prove relation is a function, one-to-one and onto. Equivalence class.

Question

Let $A$ be a nonempty set, and let $R$ be an equivalence relation on $A$. For all $a ∈ A$, let $[a]$ be the equivalence class of a on $R$. Let $r = \{(a,[a]) ∶ a ∈ A\}$.

Questions:

1. Relation $r$ is a function. True or false? Prove.
2. Relation $r$ is one-to-one. True or false? Prove.
3. Relation $r$ is onto. True or false? Prove.

Can anyone help? Thankyou.

Answer 1

It is a function $a\mapsto [a]$ with $A\to A/R$, where $A/R$ is the set of equivalence classes also called the quotient set.

It is onto because each equivalence class has pre-image, for if $X\in A/R$ then $X$ is a class, say $X=[b]$ for some $b\in A$, then $b\mapsto [b]=X$.