Answer in Discrete Mathematics for Johnson #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
Let “(a,b)in Tcirc R”. Then there exists “xin A” such that “(a,x)in T” and “(x,b)in R”. Since “Ru2286S”, we conclude that “(x,b)in S”, and therefore, “(a,b)in Tcirc S.” Consequently, “Tcirc Ru2286Tcirc S.”
Let “(a,b)in Rcirc T”. Then there exists “xin A” such that “(a,x)in R” and “(x,b)in T”. Since “Ru2286S”, we conclude that “(a,x)in S”, and therefore, “(a,b)in Scirc T.” Consequently, “Rcirc Tu2286Scirc T.”