5 Topology Questions (Including: de Morgan’s Laws)

1. Prove the following de Morgan’s laws:
(a) …
(b) …
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
(a)…
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.

Please see the attached file for the fully formatted problems.