Question

 If R, S and T are relations over the set A, then: Prove that If R⊆S, then T∘R ⊆ T∘S and R∘T ⊆ S∘T

Answer 1

Let $(a,b)\in T\circ R$. Then there exists $x\in A$ such that $(a,x)\in T$ and $(x,b)\in R$. Since $R⊆S$, we conclude that $(x,b)\in S$, and therefore, $(a,b)\in T\circ S.$ Consequently, $T\circ R⊆T\circ S.$

Let $(a,b)\in R\circ T$. Then there exists $x\in A$ such that $(a,x)\in R$ and $(x,b)\in T$. Since $R⊆S$, we conclude that $(a,x)\in S$, and therefore, $(a,b)\in S\circ T.$ Consequently, $R\circ T⊆S\circ T.$