The members of the Gainesville Film Critics Society were surveyed regarding their
cinematic heroes. 38 admire Moe 36 admire Larry 21 admire Curly 23 admire Moe and
admire Larry 5 admire Larry and admire Curly 7 admire Moe and admire Curly 3 admire
Moe and admire Larry and admire Curly 16 don't admire Moe and don't admire Larry and
don't admire Curly A. How many were surveyed? B. How many admire at least one of
the Stooges? C. How many admire Moe or Larry?
Answers
SOLUTION
GIVEN
- The members of the Gainesville Film Critics Society were surveyed regarding their cinematic heroes.
- 38 admire Moe 36 admire Larry 21 admire Curly 23 admire Moe and admire Larry 5 admire Larry and admire Curly 7 admire Moe and admire Curly 3 admire Moe and admire Larry and admire Curly 16 don't admire Moe and don't admire Larry and don't admire Curly
TO DETERMINE
A. How many were surveyed ?
B. How many admire at least one of the Stooges ?
C. How many admire Moe or Larry ?
EVALUATION
Let
A = The set of persons who admire Moe
B = The set of persons who admire Lary
C = The set of persons who admire Curly
By the given condition -
n(A) = 38
n(B) = 36
n(C) = 21
n(A ∩ B) = 23
n(A ∩ C) = 5
n(B ∩ C) = 7
n(A ∩ B ∩ C) = 3
n(A' ∩ B' ∩ C') = 16
(i) We are aware of the formula on set theory that
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C)
⇒ n(A ∪ B ∪ C) = 38 + 36 + 21 - 23 - 5 - 7 + 3
⇒ n(A ∪ B ∪ C) = 63
Hence the required number of persons surveyed
= n(A ∪ B ∪ C) + n(A' ∩ B' ∩ C')
= 63 + 16
= 79
(ii) The number of persons who admire at least one of the Stooges
= n(A ∪ B ∪ C)
= 63
(iii) The number of persons who admire Moe or Larry
= n(A ∪ B )
= n(A) + n(B) − n(A ∩ B)
= 38 + 36 - 23
= 51
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