Proof that for binary relation $ R \circ (S \cup T) \subseteq (R \circ S) \cup (R \circ T) $

Question

Proof that for binary relation following is true

$$ R \circ (S \cup T) \subseteq (R \circ S) \cup (R \circ T) $$

Attempt to proof

let $\langle x , y \rangle \in R \circ (S \cup T)$

$$ \iff \exists y : \langle x , y \rangle \in R \wedge \langle y, z \rangle \in S \cup T $$ $$ \iff \langle y , z \rangle \in S \vee \langle y ,z \rangle \in T $$ $$ \iff \langle x , z \rangle \in R \circ S \vee \langle x , z \rangle \in R \circ T $$ $$ \iff \langle x , z \rangle \in (R \circ S) \cup ( R \circ T) $$

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Is my proof correct?

Answer 1

It's not really clear what you mean by $\Leftrightarrow$.

Suppose $\langle x,y\rangle\in R\circ(S\cup T)$. Then there exists $z$ such that $$ \langle x,z\rangle\in R\quad\text{and}\quad\langle z,y\rangle\in S\cup T $$ If $\langle z,y\rangle\in S$, then $\langle x,y\rangle\in R\circ S$; if $\langle z,y\rangle\in T$, then $\langle x,y\rangle\in R\circ T$.

Therefore $\langle x,y\rangle\in (R\circ S)\cup(R\circ T)$.