(Functions preserve inclusion of sets.) Let f : X + Y be a function. Let A and A' be subsets of X such that A C A' C X,

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Functions Preserve Inclusion Of Sets Let F X Y Be A Function Let A And A Be Subsets Of X Such That A C A C X 1 (151.17 KiB) Viewed 29 times

(Functions preserve inclusion of sets.) Let f : X + Y be a function. Let A and A' be subsets of X such that A C A' C X, and let B and B' be subsets of Y such that B C B' CY. The image of A under f is defined by f(A) = {y E Y : there exists an a € A such that y = f(a)}. The inverse image of B under f is defined by /-(1) - {1 €X:/() E B}. : (a) Make a sketch showing the sets X, A, A', Y, B, B', for intuition. (b) Prove that f(A) = f(A'). (c) Prove that f-'(B) Cf-l(B'). Hint: To show that one set is contained in another set, take an arbitrary element (give it a name!) of the first set, and show that it must also belong to the second set. Use the definitions of the two sets to do so.