(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...
Question 1 Report
(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 \((A \cap B) \cup (A \cap C) = A \cap (B \cup C)\).
Answer Details
(a) With \(\varepsilon=\{1,2,\dots,20\}\):
- (i) \(A=\{\text{multiples of }5\}=\{5,10,15,20\}.\)
- (ii) \(B=\{\text{multiples of }4\}=\{4,8,12,16,20\}.\)
- (iii) \(C=\{\text{multiples of }3\}=\{3,6,9,12,15,18\}.\)
(b)
- (i) \(A\cap B=\{20\}.\)
- (ii) \(A\cap C=\{15\}.\)
- (iii) \(B\cup C=\{3,4,6,8,9,12,15,16,18,20\}.\)
(c) \[(A\cap B)\cup(A\cap C)=\{20\}\cup\{15\}=\{15,20\}.\] Also \(A\cap(B\cup C)\): from \(A=\{5,10,15,20\}\), those in \(B\cup C\) are \(\{15,20\}\). Since both sides equal \(\{15,20\}\), the identity \((A\cap B)\cup(A\cap C)=A\cap(B\cup C)\) is verified.
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