Theory question 111
Sets and LogicIn a study to find the number of languages 50 people can speak, it was revealed that (m2 + 3) of them speak Yoruba, (5m − 3) speak Hausa and (3m + 1) speak Igbo. 4 of them speak all three languages and 14 of them speak exactly two languages. Those who speak Igbo and Yoruba only are half of those who speak Yoruba and Hausa only. Those who speak Hausa and Igbo only are 1 less than those who speak Yoruba and Hausa only. 6 of them could not speak any of the languages. (a) Illustrate the information provided in a Venn Diagram (b) Find the: (i) Value of m
(ii) Number of people who speak one language only
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Let Yoruba and Hausa only = a; Igbo and Yoruba only = a/2; Hausa and Igbo only = a − 1.
a + a/2 + a − 1 = 14, so a = 6: Y∩H only 6, Y∩I only 3, H∩I only 5, all three 4, none 6.
(a) Venn diagram: three overlapping circles in a rectangle (50) with these regions; Yoruba only m² − 10, Hausa only 5m − 18, Igbo only 3m − 11, outside 6.
(b)(i) (m² − 10) + (5m − 18) + (3m − 11) + 6 + 3 + 5 + 4 + 6 = 50, so m² + 8m − 65 = 0, (m + 13)(m − 5) = 0; m = 5 (m > 0).
(ii) One language only = 15 + 7 + 4 = 26.
Find the 'exactly two' regions first from the given relations.