(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) \((A \cap B) \cup (A \cap C) = A \cap (B \cup C)\) \((A \cap B) \cup (A \cap C) = {15, 20}\) \(A \cap (B \cup C) = {5, 10, 15, 20} \cap {3, 4, 6, 8, 9, 12, 15, 16, 18, 20}\) = \({15, 20}\) \(\therefore (A \cap B) \cup (A \cap C) = A \cap (B \cup C)\)
