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Let A = \{20, 25, 26, 30, 31, 33, 35\}, B = \{25, 30, 32, 34, 35, 36\} and C = \{15, 26, 30, 32, 33, 35\}.
(i) Calculate the total number of common elements in all the three sets.
(ii) Calculate the total number of common elements in A and B.
(iii) Calculate the total number of common elements in B and C.
(iv) Calculate the total number of common elements in A and C.
(v) Calculate the total number of elements in all the three sets.
Solution:
A = \{20, 25, 26, 30, 31, 33, 35\}
Number of elements in set A = n(A) = 7
B = \{25, 30, 32, 34, 35, 36\}
Number of elements in set B = n(B) = 6
C = \{15, 26, 30, 32, 33, 35\}
Number of elements in set C = n(C) = 6
(i) Total number of common elements in all the three sets:
A \cap B \cap C = \{30, 35\}
n(A \cap B \cap C) = 2
(ii) Total number of common elements in A and B:
A \cap B = \{25, 30, 35\}
n(A \cap B) = 3
(iii) Total number of common elements in B and C:
B \cap C = \{30, 32, 35\}
n(B \cap C) = 3
(iv) Total number of common elements in A and C:
A \cap C = \{26, 30, 33, 35\}
n(A \cap C) = 4
(v) Total number of elements in all the three sets:
n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)
n(A \cup B \cup C) = 7 + 6 + 6 - 3 - 3 - 4 + 2
n(A \cup B \cup C) = 11
Important!
If A and B are two finite sets, then:
1. n(A \cup B) = n(A) + n(B) - n(A \cap B)
2. n(A - B) = n(A) - n(A \cap B)
3. n(B - A) = n(B) - n(A \cap B)
4. n(A^{\prime}) = n(U) - n(A)
5. n(U) = n(A) + n(A^{\prime})
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