1. Prove the following de Morgan's laws:
2. Let A be a set. For each p E A, let Gp be a subset of A such that p C Gp C A. Then show that A = Up E A Gp.
3. Let f : X ---> Y be a function and A, B C Y. Then show that
4. Let f : X ?> Y be a function and A C X, B C V. Then show that
(a) A C f-1 o f(A).
(b) B = f o f-1(B).
5. Let f X ?> and g : V ?> Z. Prove that
(a) if f and g are onto, then y o f : X ?'--* Z is onto.
(b) if f and g are one-to-one, then g o f is one-to-one.
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a. For any , we have , then for any . This implies that for any . So . On the other hand, if , then for any . So for any . This implies that . Thus . Therefore, .
b. For any , we have , then for some . This implies that for some . So . On the other hand, if we have , then for some . This means that for some . So . This implies that . ...
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