Let f : X→Y Prove f ^-1 (C-D) is a sub set of f ^-1 (C) – f ^-1 (D)?

Question

Let f : X→Y 
Prove f -1 (C-D) is a sub set of  f -1 (C) – f -1 (D)

mcbengt · Accepted Answer

Let x be an arbitrary element of f^(-1)(C-D).  By definition of f^(-1)(C-D), this means that

f(x) is in C - D.

By definition of C - D this means that

f(x) is in C, and f(x) is not in D.

By definition of membership in f^(-1)(C) we conclude that x is in f^(-1)(C), and by definition of membership in f^(-1)(D) we conclude that x is not in f^(-1)(D).

Thus by definition of membership in f^(-1)(C) - f^(-1)(D) we conclude that x is in f^(-1)(C) - f^(-1)(D).

Since x was an arbitrary element of f^(-1)(C-D) we conclude that f^(-1)(C-D) is a subset of f^(-1)(C) - f^(-1)(D).

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Let f : X→Y Prove f ^-1 (C-D) is a sub set of f ^-1 (C) – f ^-1 (D)?

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