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

Question

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Answer 1

Proof:

1. Suppose (a,b) ∈ T ο R . Then there exists c ∈ A such that (a,c) ∈ R and (c,b) ∈ T. Since R ⊆ S, (a,c) ∈ S and thus (a,b) ∈ T ο S. Therefore, T ο R ⊆ T ο S.
2. Suppose (a,b) ∈ R ο T. Then there exists c ∈ A such that (a,c) ∈ T and (c,b) ∈ R. Since R ⊆ S, (c,b) ∈ S and thus (a,b) ∈ S ο T. Therefore, R ο T ⊆ S ο T.