Determine whether the following statements are true or false: (a) The empty set ∅ is a subset of {1, 2, 3}. (b) If A = {1, 2, 3}, then {1} is a subset of ℘(A).

Let U= {1, 2, …9} be the universal set and let A= {1, 2 , 3, 4, 5} and C= {5, 6, 7, 8, 9} and B= {1, 3, 4, 5} Find the set. 1. (A \ B')'

prove: for any two sets X and Y, #(X x Y) = #(X) • #(Y)

Let A and B be infinite sets that are countable. Do A and B necessarily have the same cardinality as each other? If yes explain. If no give a counter example

(4) Let A, B, C be subsets of a universal set U. Determine if the following statements are true or false. If true, prove them. If false, find a counterexample. (a) An(B\C) = (A \ B) n (A \ C) (b) If Ac C BC, then AC B. (c) (ANB) UC = (AUC) n(BUC).

Let A be a set and define A = {1,2, x,y}. So which of the following is A2 ?

Let A = {5,6, 7}. What is P(A)? Do not merely describe the set, but rather write down all the elements.

use an element of argument to prove for all sets A,B,C (A-B)∩(C-B) = (A∩C)-B please explain if possible

2. Let A = {0,2,4, 6,8} and B = {1,3,5,7} have universal set U = {0,1,2,...,8}. Find: (a) Ā (b) В (d) AUĀ (g) ĀnĒ (h) AnB (е) А -А (c) AnĀ (f) AUB (i) AxB

Give an example of an infinite collection of open sets whose intersection is not open.