If R, S and T are relations over the set A, then: Prove that (S∩T)∘R= (S∘R)∩(T∘R).
Let x and y be the arbitary element belongs to set A,
(x,y)∈R,S,T(x,y)\in R,S,T(x,y)∈R,S,T
(x,y)∈(S∩T)oR(x,y)\in(S\cap T)oR(x,y)∈(S∩T)oR
⇒x∈(SoR)∩(ToR)( As It follows Distributive law)⇒x\in (So R)\cap(ToR) (\text{ As It follows Distributive law})⇒x∈(SoR)∩(ToR)( As It follows Distributive law)
⇒(S∩R)oR=(SoR)∩(ToR)\Rightarrow (S\cap R)oR=(SoR)\cap(ToR)⇒(S∩R)oR=(SoR)∩(ToR) ,hence proved.
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