(a) If (varepsilon) is the set ({1, 2, 3,…, 19, 20}) and A, B and C are subsets of (varepsilon) such that A = { multiples of five}, B = {multiples of four} and C = {multiples of three}, list the elements of (i) A ; (ii) B ; (iii) C ;
(b) Find : (i) (A cap B) ; (ii) (A cap C) ; (iii) (B cup C).
(c) Using your results in (b), show that ((A cap B) cup (A cap C) = A cap (B cup C)).
Explanation
(a)(i) A = {5, 10, 15, 20}
(ii) B = {4, 8, 12, 16, 20}
(iii) C = {3, 6, 9, 12, 15, 18}
(b) (i) (A cap B = {20})
(ii) (A cap C = {15})
(iii) (B cap C = {12})
(c) ((A cap B) cup (A cap C) = A cap (B cup C))
((A cap B) cup (A cap C) = {15, 20})
(A cap (B cup C) = {5, 10, 15, 20} cap {3, 4, 6, 8, 9, 12, 15, 16, 18, 20})
= ({15, 20})
(therefore (A cap B) cup (A cap C) = A cap (B cup C))