# 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.

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

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Intersecting Lines | What Are Intersecting Lines? | Definition

Jun 14, 24 11:00 AM

Two lines that cross each other at a particular point are called intersecting lines. The point where two lines cross is called the point of intersection. In the given figure AB and CD intersect each o…

2. ### Line-Segment, Ray and Line | Definition of in Line-segment | Symbol

Jun 14, 24 10:41 AM

Definition of in Line-segment, ray and line geometry: A line segment is a fixed part of a line. It has two end points. It is named by the end points. In the figure given below end points are A and B…

3. ### Definition of Points, Lines and Shapes in Geometry | Types & Examples

Jun 14, 24 09:45 AM

Definition of points, lines and shapes in geometry: Point: A point is the fundamental element of geometry. If we put the tip of a pencil on a paper and press it lightly,

4. ### Subtracting Integers | Subtraction of Integers |Fundamental Operations

Jun 13, 24 04:32 PM

Subtracting integers is the second operations on integers, among the four fundamental operations on integers. Change the sign of the integer to be subtracted and then add.