Disjoint sets:
If two sets A and B have no similar members i.e. if no component of A is in B and no element of B is in A, then A and B are called be Disjoint Sets. Therefore for Disjoint Sets A and B => n (A ∩ B) = 0.
Some More Results related to the Order of Finite Sets:
Take A, B and C be finite sets and U be the finite universal set, then
(i). n (A ∪ B) = n (A) + n (B) - n (A ∩ B)
(ii). If A and B are disjoint, then n (A ∪ B) = n (A) + n (B)
(iii). n (A -B) = n (A) - n (A ∩ B) i.e. n (A) = n (A - B) + n (A ∩ B)
(iv). n (A ∪ B ∪ C) = n (A) + n (B) + n (C) - n (A ∩ B) - n (B ∩ C) - n (A ∩ C) + n (A ∩ B ∩ C)
(v). n (set of elements which are in exactly two of the sets A, B, C)
= n (A ∩ B)+n (B ∩ C) + n (C ∩ A) -3n(A ∩ B ∩ C)
(vi). n(set of elements which are in at least two of the sets A, B, C)
= n (A ∩ B) + n (A ∩ C) + n (B ∩ C) -2n(A ∩ B ∩ C)
(vii). n (set of elements which are in exactly one of the sets A, B, C)
= n (A) + n (B) + n (C) - 2n (A ∩ B) - 2n (B ∩ C) - 2n (A ∩ C) + 3n (A ∩ B ∩ C)
Explanation: If A and B be two sets having 3 and 6 members respectively, what may be the minimum number of components in A È B? search also, the maximum number of members in A È B.
Solution: We have, n (A ∪ B) = n(A) + n(B) - n(A ∩ B)
That defines that n (A ∪ B) is maximum or minimum related as
n (A ∩ B) is maximum or minimum respectively.
Case 1: When n (A ∩ B) is minimum, ie. n (A ∩ B) = 0. That is possible only when A ∩ B = f. In this case,
n(A ∪ B) = n (A) + n (B) - 0 = n(A) + n (B) = 3 +6 = 9
n (A ∪ B)max = 9
Case 2: When n (A ∩ B) is maximum
That is right only when A ⊆ B.
In this form n (A ∩ B) = 3
∴ n (A ∪ B) = n(A) + n(B) - n (A∩B) = (3+6-3)=6
n (A ∪ B)min = 6.
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