Suppose that each of A, B, and C is a set and each of BnC, A–(BnC), A-B, A-C, and (A – B) n (A – C) has at least one member. Does it follow that A – (BnC) = (A – B) n (A –- C)? If so, prove that it does. If not, give a specific example to show that it does not.

(a)Calculate A ⊕ B ⊕ C for A = {1, 2, 3, 5}, B = {1, 2, 4, 6}, C = {1, 3, 4, 7}.Note that the symmetric difference operation is associative: (A ⊕ B) ⊕ C = A ⊕ (B ⊕ C).(b)Let A, B, and C be any finite sets. Give a concise description for which elements from A, B, and C are in A ⊕ B ⊕ C.(c)Let A1, A2, …, An be any finite sets. Give a concise description for which elements from A1, A2, …, An are in A1 ⊕ A2 ⊕ … ⊕ An.

When |S| denotes the number of elements of the set S, and S = {x/1 ≤ x ≤ 100, x € integer } There are three subsets of S, namely sets A, B and C. It is known |A|=(10+k), |B|= (7+2k), |C|= (8+2k), |A∩B|=(2+k), |A∩C|=(3 +k), |B∩C|=2k, |A∩B∩C|=k and |(AUBUC)'|=40 Draw the Venn diagram, determine the value of k ? Determine the number of elements of each set A, B and C?

Exercise 5. Prove that given positive integers a and b, there is a positive integer d such that • d divides a and d divides b, and • if c is a positive integer which divides both a and b then c divides d (in other words, any common divisor of a and b must divide d). HINT: Start this proof by defining a set D {ax + by | x, y E Z and a set C = D N Z,o. What