# Operations on Sets

Operations on sets include intersection, union and difference of two sets.

1) If A = { x | x ≤6 ; x ∈ N} and B = {x |3 ≤ x < 9, x ∈ N} ,

find i) A ∪ B (ii) A ∩ B (iii) A - B (iv) B - A.

A = { x | x ≤6 ; x ∈ N} = { 1,2,3,4,5,6}

B = {x |3 ≤ x < 9, x ∈ N} = { 3,4,5,6,7,8}

(i) A ∪ B = { 1,2,3,4,5,6} ∪ { 3,4,5,6,7,8}

(ii) A ∩ B = { 1,2,3,4,5,6} ∩ { 3,4,5,6,7,8}

(iii) A - B = { 1,2,3,4,5,6} - { 3,4,5,6,7,8}

(iv) B - A = { 3,4,5,6,7,8} - { 1,2,3,4,5,6}

_______________________________________________________________

2) If U = ξ = {x |x ≤ 8, x ∈ W}, A = { x|2x -1 ≤ 8; x ∈ W}and

B = { x|5x -1 ≤ 14, x ∈ N}. Find (i) (A ∪ B)' (ii) A' ∩ B'

U = ξ = {x |x ≤ 8, x ∈ W} = {0,1,2,3,4,5,6,7,8}

A = { x|2x -1 ≤ 8; x ∈ W}= { x|x ≤ 9/2; x ∈ W} = {0,1,2,3,4}

B = { x|5x -1 ≤ 14, x ∈ N}= {x|x ≤ 3, x ∈ N} ={1,2,3}

(i) (A ∪ B)= {0,1,2,3,4} ∪ {1,2,3}= {0,1,2,3,4}

∴

(ii) A' = {5,6,7,8} and B' = {0,4,5,6,7,8}

∴

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3) If A ={ 6,7,8,9,10,11,12,13,14} , B= { 11,12,13,14,15,16,17,18,19} and

C = {16,17,18,19,20,21,22,23,24}. Find A ∪ ( B ∩ C)

First we will find B ∩ C

B ∩ C = { 11,12,13,14,15,16,17,18,19} ∩ {16,17,18,19,20,21,22,23,24}

B ∩ C = { 16,17,18,19}

Now, A ∪ ( B ∩ C) = { 6,7,8,9,10,11,12,13,14} ∪ { 16,17,18,19}

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**Examples on operations on sets :**1) If A = { x | x ≤6 ; x ∈ N} and B = {x |3 ≤ x < 9, x ∈ N} ,

find i) A ∪ B (ii) A ∩ B (iii) A - B (iv) B - A.

**Solution :**A = { x | x ≤6 ; x ∈ N} = { 1,2,3,4,5,6}

B = {x |3 ≤ x < 9, x ∈ N} = { 3,4,5,6,7,8}

(i) A ∪ B = { 1,2,3,4,5,6} ∪ { 3,4,5,6,7,8}

**= { 1,2,3,4,5,6,7,8}**.(ii) A ∩ B = { 1,2,3,4,5,6} ∩ { 3,4,5,6,7,8}

**= {3,4,5,6}**.(iii) A - B = { 1,2,3,4,5,6} - { 3,4,5,6,7,8}

**= {1,2}**(iv) B - A = { 3,4,5,6,7,8} - { 1,2,3,4,5,6}

**= { 7,8}**_______________________________________________________________

2) If U = ξ = {x |x ≤ 8, x ∈ W}, A = { x|2x -1 ≤ 8; x ∈ W}and

B = { x|5x -1 ≤ 14, x ∈ N}. Find (i) (A ∪ B)' (ii) A' ∩ B'

**Solution :**U = ξ = {x |x ≤ 8, x ∈ W} = {0,1,2,3,4,5,6,7,8}

A = { x|2x -1 ≤ 8; x ∈ W}= { x|x ≤ 9/2; x ∈ W} = {0,1,2,3,4}

B = { x|5x -1 ≤ 14, x ∈ N}= {x|x ≤ 3, x ∈ N} ={1,2,3}

(i) (A ∪ B)= {0,1,2,3,4} ∪ {1,2,3}= {0,1,2,3,4}

∴

**(A ∪ B)' = {5,6,7,8}**(ii) A' = {5,6,7,8} and B' = {0,4,5,6,7,8}

∴

**A' ∩ B'= {5,6,7,8}**_______________________________________________________________

3) If A ={ 6,7,8,9,10,11,12,13,14} , B= { 11,12,13,14,15,16,17,18,19} and

C = {16,17,18,19,20,21,22,23,24}. Find A ∪ ( B ∩ C)

**Solution:**First we will find B ∩ C

B ∩ C = { 11,12,13,14,15,16,17,18,19} ∩ {16,17,18,19,20,21,22,23,24}

B ∩ C = { 16,17,18,19}

Now, A ∪ ( B ∩ C) = { 6,7,8,9,10,11,12,13,14} ∪ { 16,17,18,19}

**A ∪ ( B ∩ C) = {6,7,8,9,10,11,12,13,14, 16,17,18,19}****Set Theory**

• Sets

• Representation of Set

• Cardinal Number

• Types of Sets

• Pairs of Sets

• Subset

• Complement of Set

• Union of the Sets

• Intersection of Sets

• Operations on Sets

• De Morgan's Law

• Venn Diagrams

• Venn-diagrams for sets

• Venn-diagrams for different situations

• Problems on Intersection of Two Sets

• Problems on Intersection of Three Sets

• Sets

• Representation of Set

• Cardinal Number

• Types of Sets

• Pairs of Sets

• Subset

• Complement of Set

• Union of the Sets

• Intersection of Sets

• Operations on Sets

• De Morgan's Law

• Venn Diagrams

• Venn-diagrams for sets

• Venn-diagrams for different situations

• Problems on Intersection of Two Sets

• Problems on Intersection of Three Sets