Set Theory Practice Solution3





In this page set theory practice solution3 we are going to see solution of practice questions from the worksheet set theory practice questions1.

Question 9

For A = {5,10,15,20} B = {6,10,12,18,24} and C ={7,10,12,14,21,28} verify whether A\ (B\C) = (A\B)\C. Justify your answer.

Solution:

A = {5,10,15,20}

B = {6,10,12,18,24}

C = {7,10,12,14,21,28}

L.H.S

A\(B\C)

(B\C) = {6,10,12,18,24} \ {7,10,12,14,21,28}

        = {6,18,24}

A\(B\C) = {5,10,15,20} \ {6,18,24}

           = {5,10,15,20}  ---(1)

(A\B) = {5,10,15,20}\{6,10,12,18,24}

        = {5,15,20}

(A\B)\C = {5,15,20} \ {7,10,12,14,21,28}

           = {5,15,20}   --- (2)

A\ (B\C) ≠ (A\B)\C


Question 10

Let A = {-5,-3,-2,-1} B = {-2,-1,0} and C = {-6,-4,-2}. Find A\(B\C) and (A\B)\C. What can we conclude about set difference operation?

Solution:

A = {-5,-3,-2,-1} B = {-2,-1,0} and C = {-6,-4,-2}

L.H.S

A\(B\C)

(B\C) = {-2,-1,0} \ {-6,-4,-2}

        = {-1,0}

A\(B\C) = {-5,-3,-2,-1} \ {-1,0}

           = {-5,-3,-2} ---- (1)

R.H.S

(A\B)\C

(A\B) = {-5,-3,-2,-1} \ {-2,-1,0}

        = {-5,-3}

(A\B)\C = {-5,-3} \ {-6,-4,-2}

           = {-5,-3}  -----(2)

(1) ≠ (2)

A\(B\C) ≠ (A\B)\C


(11) For A = {-3,-1,0,4,6,8,10} B = {-1,-2,3,4,5,6} and C = {-1,2,3,4,5,7},show that

(i) A U (B ∩ C) = (A U B) ∩ (A U C)

(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

(iii) Verify using Venn diagrams

Solution:

(i) A U (B ∩ C) = (A U B) ∩ (A U C)

L.H.S

A U (B ∩ C)

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

           = {-1,3,4,5}

A U (B ∩ C) = {-3,-1,0,4,6,8,10} U {-1,3,4,5}

                = {-3,-1,0,3,4,5,6,8,10}  --- (1)

R.H.S

(A U B) ∩ (A U C)

 (A U B) = {-3,-1,0,4,6,8,10} U {-1,-2,3,4,5,6}

            = {-3,-2,-1,0,3,4,5,6,8,10}

 (A U C) = {-3,-1,0,4,6,8,10} U {-1,2,3,4,5,7}

           = {-3,-1,0,2,3,4,5,6,7,8,10}

(AUB)∩(AUC)={-3,-2,-1,0,3,4,5,6,8,10} ∩ {-3,-1,0,2,3,4,5,6,7,8,10}

                 = {-3,-1,0,3,4,5,6,8,10}  --- (2)

(1) = (2)

A U (B ∩ C) = (A U B) ∩ (A U C)

Hence proved


(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

A = {-3,-1,0,4,6,8,10} B = {-1,-2,3,4,5,6} and C = {-1,2,3,4,5,7}

L.H.S

(B U C) = {-1,-2,3,4,5,6} U {-1,2,3,4,5,7}

           = {-2,-1,2,3,4,5,6,7}

A ∩ (B U C) = {-3,-1,0,4,6,8,10} ∩ {-2,-1,2,3,4,5,6,7}

                = {-1,4,6} ---- (1)

R.H.S

(A ∩ B) = {-3,-1,0,4,6,8,10} ∩ {-1,-2,3,4,5,6}

           = {-1,4,6}

(A ∩ C) = {-3,-1,0,4,6,8,10} ∩ {-1,2,3,4,5,7}

           = {-1,4}

(A ∩ B) U (A ∩ C) = {-1,4,6} U {-1,4}

                        = {-1,4,6} ---- (2)

 (1) = (2)

A ∩ (B U C) = (A ∩ B) U (A ∩ C)

Hence proved

set theory practice solution3  set theory practice solution3