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 βˆͺ (B ∩ C) = (A βˆͺ B) ∩ (A βˆͺ C)

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

Solution:

(i) A βˆͺ (B ∩ C) = (A βˆͺ B) ∩ (A βˆͺ C)

L.H.S. = A βˆͺ (B ∩ C)

B ∩ C = {3}

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

R.H.S. = (A βˆͺ B) ∩ (A βˆͺ C)

A βˆͺ B = {1, 3, 5, 6}

A βˆͺ C = {1, 3, 5, 7}

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

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

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

(ii) A ∩ (B βˆͺ C) = (A ∩ B) βˆͺ (A ∩ C)

L.H.S. = A ∩ (B βˆͺ C)

B βˆͺ C = {1, 3, 5, 6, 7}

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

A ∩ B = {3, 5}

A ∩ C = {1, 3}

(A ∩ B) βˆͺ (A ∩ C) = {3, 5} βˆͺ {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 βˆͺ C) = (A ∩ B) βˆͺ (A ∩ C)

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

Solution:

(i) A ∩ (B βˆͺ C) = (A ∩ B) βˆͺ (A ∩ C)

L.H.S. = A ∩ (B βˆͺ C)

B βˆͺ C = {b, c, d, e, f, g}

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

R.H.S. = (A ∩ B) βˆͺ (A ∩ C)

A ∩ B = {b, e}

A ∩ C = {d, e}

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

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

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

(ii) A βˆͺ (B ∩ C) = (A βˆͺ B) ∩ (A βˆͺ C)

L.H.S. = A βˆͺ (B ∩ C)

B ∩ C = {e, f}

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

R.H.S. = (A βˆͺ B) ∩ (A βˆͺ C)

AβˆͺB = {a, b, c, d, e, f}

AβˆͺC = {a, b, d, e, f, g}

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

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

A βˆͺ (B ∩ C) = A βˆͺ B ∩ (A βˆͺ 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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