why $(\theta _{1}\circ \theta _{2})^{\vee }\subset(\theta _{2}\circ \theta _{1})^{\vee } $

Question

my questiones at this theorem: i coud not undrestand $a\Rightarrow b$ and why $(\theta _{1}\circ \theta _{2})^{\vee }\subset(\theta _{2}\circ \theta _{1})^{\vee } $ please guide me?

Answer 1

Do you know the definition of the relational inverse operation? Assuming you do, then for binary relations $R$ and $S$, verify the following: if $R \subseteq S$, then $R^{\vee} \subseteq S^{\vee}$. Then check that $(\theta_1\circ \theta_2)^{\vee} = \theta_2^{\vee}\circ \theta_1^{\vee}$.