Answer to Question #179899 in Discrete Mathematics for Johnson

Question #179899

 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


1
Expert's answer
2021-04-28T16:09:58-0400

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\u2286S", we conclude that "(x,b)\\in S", and therefore, "(a,b)\\in T\\circ S." Consequently, "T\\circ R\u2286T\\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\u2286S", we conclude that "(a,x)\\in S", and therefore, "(a,b)\\in S\\circ T." Consequently, "R\\circ T\u2286S\\circ T."

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