To prove or give counterexample that $R \subset Dom(R) \times Ran(R)$

Question

To prove or give counterexample that $R \subset Dom(R) \times Ran(R)$

Let (a,b) \in R. So $a \in A$ and $b \in B$ where $R :A \rightarrow B$. Since Dom(R) is asubset of A. Let assume that $a \not \in Dom(R) $. So (a,b) doesnot belong to $ Dom(R) \times Ran(R)$

Is this correct ? Thanks for help

Answer 1

If $(a,b)\in R$ then $a\in Dom(R)$ by the definition of "domain" and $b \in Ran(R)$ by the definition of "range", so $(a,b)\in Dom(R) \times Ran(R)$, so $R \subseteq Dom(R) \times Ran(R)$. QED

Answer 2

I can see that you're trying to use a proof by contradiction, but it isn't very clear what you're trying to contradict. It's easier to use a direct proof, and it would look something like this:

Proof. Let $R:A\rightarrow B$ be a mapping from $A$ to $B$ where $R=\{a\in A, b\in B: R(a)=b\}$.
Let $(a,b)\in R$. Then by definition, $R(a)=b$, so $a$ is in the domain of $R$ and $b$ is in the range of $R$. So $(a,b)\in$ Dom$(R)\times$Dom$(R)$. Therefore $R\subseteq$ Dom$(R)\times$Dom$(R)$.