If Phi D Rightarrow R and P and Q are subsets of R prove tha

If Phi: D Rightarrow R and P and Q are subsets of R, prove that Phi^-1 (P Intersection Q) =Phi^-1 (P) Intersection Phi^-1(Q).

Solution

*We know that if f: X Y and if A B in X , then f^- (A) f^- (B) in Y.

Therefore, Ø^-1(PQ) Ø^-1(P) and Ø^-1(PQ) Ø^-1 (Q) (as P Q P and P Q Q) . Thus, Ø^-1 (P Q) Ø^-1 (P) Ø^-1 (Q) …(1)

Let x Ø^-1(P) Ø^-1 (Q). Then Ø (x) P and Ø (x) Q. Therefore, Ø (x) P Q. Thus x Ø^-1 (P Q).

Therefore, Ø^-1(P) Ø^-1 (Q) Ø^-1 (P B)…(2)

Hence , Ø^-1(A) Ø^-1 (B) = Ø^-1 (A Q).

Note: *

Let f(x) f(A). Then x A. Since A B, therefore x B so that f(x) f(B). Therefore, f(A) f(B).