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Set Theory

Probability Introduction :

Sample Questions

1Q) Let A = {1, 2, 3}, B = {2, 3, 4}, C = {4, 5, 6, 7} and U = {1, 2, 3, 4, 5, 6, 7}.

a) A ∩ B, B ∩ A    

Solution:-

A ∩ B = {1, 2, 3} ∩ {2, 3, 4}

           = {2, 3}

B ∩ A = {2, 3, 4} ∩ {1, 2, 3}

           = {2, 3}

b) (A ∪ B) ∪ C , A ∪ (B ∪ C)

Solution:-

   A ∪ B = {1, 2, 3} ∪ {2, 3, 4}

              = {1, 2, 3, 4}

   (A ∪ B) ∪ C = {1, 2, 3, 4} ∪ {4, 5, 6, 7}

                        = {1, 2, 3, 4, 5, 6, 7}

   B ∪ C = {2, 3, 4} ∪ {4, 5, 6, 7}

              = {2, 3, 4, 5, 6, 7}

   A ∪ (B ∪ C) = {1, 2, 3} ∪ {2, 3, 4, 5, 6, 7}

                        = {1, 2, 3, 4, 5, 6, 7}

c) A ∪ B, B ∪ A

Solution:-

   A ∪ B = {1, 2, 3} ∪ {2, 3, 4}

              = { 1, 2, 3, 4}

   B ∪ A = {2, 3, 4} ∪ {1, 2, 3}

              = {1, 2, 3, 4}

d) (A ∩ B) ∩ C ,  A ∩ (B ∩ C)

Solution:-

   A ∩ B = {1, 2, 3} ∩ {2, 3, 4}

                = {2, 3}

   (A ∩ B) ∩ C = {2, 3} ∩ {4, 5, 6, 7}

                           = ∅

   B ∩ C = {2, 3, 4} ∩ {4, 5, 6, 7}

                = {4}

   A ∩ (B ∩ C) = {1, 2, 3} ∩ {4}

                           = ∅

e) A ∪ (B ∩ C) , (A ∪ B) ∩ (A ∪ C)

Solution:-

(B ∩ C) = {4}

A ∪ (B ∩ C) = {1, 2, 3} ∪ {4}

                         = {1, 2, 3, 4}

f) A ∩ (B ∪ C) , (A ∩ B) ∪ (A ∩ C)

   (B ∪ C) = {2, 3, 4, 5, 6, 7}

   A ∩ (B ∪ C) = {1, 2, 3} ∩ {2, 3, 4, 5, 6, 7}

                           = {2, 3}

   (A ∩ B) = {1, 2, 3} ∩ {2, 3, 4}

                   = {2, 3}

   (A ∩ C) = {1, 2, 3} ∩ {4, 5, 6, 7}

                   = ∅

   (A ∩ B) ∪ (A ∩ C) = {2, 3} ∪ ∅

                                       = {2, 3}



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