Disjoint sets Assignment Help

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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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