Set Theory
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These are the solutions to the WASSCE past questions on the topic: Sets.
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Out of the number, 24 preferred Cymbal, 16 Saxophone and 18 Bongo.
8 preferred Cymbal only, 2 Saxophone only and 6 Bongo only.
4 played all the three instruments while 7 preferred Cymbal and Saxophone only.
(a.) Illustrate the information in a Venn diagram.
(b.) Find the number of artistes who preferred:
(i.) only two types of instruments;
(ii.) only one type of instrument.
Let:
Cymbal = C
Saxophone = S
Bongo = B
universal set = μ
$ n(\mu) = 35 \\[3ex] n(C) = 24 \\[3ex] n(S) = 16 \\[3ex] n(B) = 18 \\[3ex] n(\text{only C}) = 8 \\[3ex] n(\text{only S}) = 2 \\[3ex] n(\text{only B}) = 6 \\[3ex] n(\text{C and S and B}) = 4 \\[3ex] n(\text{only C and S}) = 7 \\[3ex] n(\text{only C and B}) = m \\[3ex] n(\text{only S and B}) = p \\[3ex] $ The Venn diagram based on the information is:
$ 8 + 7 + 4 + m = n(C) \\[3ex] 19 + m = 24 \\[3ex] m = 24 - 19 \\[3ex] m = 5 \\[5ex] 7 + 2 + 4 + p = n(S) \\[3ex] 13 + p = 16 \\[3ex] p = 16 - 13 \\[3ex] p = 3 \\[5ex] 8 + 7 + 4 + 5 + 2 + 3 + 6 = 35 \\[3ex] n(\text{neither of C, S, and B}) = 35 - 35 = 0 \\[3ex] $ (a.) The updated Venn diagram is:
$ (b.)(i.) \\[3ex] n(\text{only two types of instruments}) \\[3ex] = n(\text{only C and S}) + n(\text{only C and B}) + n(\text{only S and B}) \\[3ex] = 7 + 5 + 3 \\[3ex] = 15\;artistes \\[5ex] (ii.) \\[3ex] n(\text{only one type of instrument}) \\[3ex] = n(\text{only C}) + n(\text{only S}) + n(\text{only B}) \\[3ex] = 8 + 2 + 6 \\[3ex] = 16\;artistes $
