Problems on Operation on Sets

Solved problems on operation on sets are given below to get a fair idea how to find the union and intersection of two or more sets.


We know, the union of sets is a set which contains all the elements in those sets and intersection of sets is a set which contains all the elements that are common in those sets.

Click Here to know more about the two basic operations on sets.


Solved problems on operation on sets:

1. If A = {1, 3, 5}, B = {3, 5, 6} and C = {1, 3, 7} 

(i) Verify that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

(ii) Verify A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Solution:

(i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

L.H.S. = A ∪ (B ∩ C)

B ∩ C = {3}

A ∪ (B ∩ C) = {1, 3, 5} ∪ {3} = {1, 3, 5} ……………….. (1)

R.H.S. = (A ∪ B) ∩ (A ∪ C)

A ∪ B = {1, 3, 5, 6}

A ∪ C = {1, 3, 5, 7}

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

From (1) and (2), we conclude that;

A ∪ (B ∩ C) = A ∪ B ∩ (A ∪ C)  [verified]

(ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

L.H.S. = A ∩ (B ∪ C)

B ∪ C = {1, 3, 5, 6, 7}

A ∩ (B ∪ C) = {1, 3, 5} ∩ {1, 3, 5, 6, 7} = {1, 3, 5}     ……………….. (1)
R.H.S. = (A ∩ B) ∪ (A ∩ C)

A ∩ B = {3, 5}

A ∩ C = {1, 3}

(A ∩ B) ∪ (A ∩ C) = {3, 5} ∪ {1, 3} = {1, 3, 5}     ……………….. (2)

From (1) and (2), we conclude that;

A ∩ (B ⋃ C) = (A ∩ B) ⋃ (A ∩ C)  [verified]

More worked-out problems on operation on sets to find the union and intersection of three sets.

2. Let A = {a, b, d, e}, B = {b, c, e, f} and C = {d, e, f, g}

(i) Verify A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

(ii) Verify A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Solution:

(i) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

L.H.S. = A ∩ (B ∪ C)

B ∪ C = {b, c, d, e, f, g}

A ∩ (B ∪ C) = {b, d, e}     ……………….. (1)

R.H.S. = (A ∩ B) ∪ (A ∩ C)

A ∩ B = {b, e}

A ∩ C = {d, e}

(A ∩ B) ∪ (A ∩ C) = {b, d, e}     ……………….. (2)

From (1) and (2), we conclude that;

A ∩ (B ⋃ C) = (A ∩ B) ⋃ (A ∩ C)  [verified]

(ii) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

L.H.S. = A ∪ (B ∩ C)

B ∩ C = {e, f}

A ∪ (B ∩ C) = {a, b, d, e, f}     ……………….. (1)

R.H.S. = (A ∪ B) ∩ (A ∪ C)

A∪B = {a, b, c, d, e, f}

A∪C = {a, b, d, e, f, g}

(A ∪ B) ∩ (A ∪ C) = {a, b, d, e, f}     ……………….. (2)

From (1) and (2), we conclude that;

A ∪ (B ∩ C) = A ∪ B ∩ (A ∪ C)  [verified]

Set Theory

Sets Theory

Representation of a Set

Types of Sets

Finite Sets and Infinite Sets

Power Set

Problems on Union of Sets

Problems on Intersection of Sets

Difference of two Sets

Complement of a Set

Problems on Complement of a Set

Problems on Operation on Sets

Word Problems on Sets

Venn Diagrams in Different Situations

Relationship in Sets using Venn Diagram

Union of Sets using Venn Diagram

Intersection of Sets using Venn Diagram

Disjoint of Sets using Venn Diagram

Difference of Sets using Venn Diagram

Examples on Venn Diagram








8th Grade Math Practice

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