Theory question 106
SetsIn a remedial school of 13 teachers, 8 teach Government (G), 7 teach Geography (P) and 7 teach History (H). 2 teach Government and Geography only, 2 teach Government and History only and 3 teach Geography and History only.
(a) Represent the information in a Venn diagram
(b) Find: (i) n(G ∩ P ∩ H) (ii) n(P ∩ H) (iii) n[(P ∪ H)′ ∩ G]
Show the model answer
Model answer
(a) Three overlapping circles G, P, H in U (13); G∩P only = 2, G∩H only = 2, P∩H only = 3, all three = x; G only = 4 − x, P only = 2 − x, H only = 2 − x
(b)(i) (4 − x) + (2 − x) + (2 − x) + 2 + 2 + 3 + x = 13 ⇒ 15 − 2x = 13 ⇒ x = 1
(ii) n(P ∩ H) = 3 + 1 = 4
(iii) n[(P ∪ H)′ ∩ G] = Government only = 4 − 1 = 3
Explanation
Every teacher teaches at least one subject, so all regions add up to 13.