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

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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, $R\circ T⊆S\circ T.$

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

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Question ID: mtid-5-stid-8-sqid-3190-qpid-1889