# Union of the Sets

The set consisting of all elements of the set A and set B is called the union of the sets A and B. It is denoted by A ∪ B. ( Read as A union B).

1) A ∪ A = A

2) A ∪ φ = A

3) A ∪ B = B ∪ A (Commutative property for union)

4) ( A ∪ B ) ∪ C = A ∪ ( B ∪ C ) ( Associative property )

5) U ∪ A = U (Law of U)

1) If A = { 1, 2, 3, 4} and B = { 2, 3, 5, 6, 7 } then find A ∪ B.

A = { 1, 2, 3, 4} and B = { 2, 3, 5, 6, 7 }

A union B is obtained by combining the two sets but if there is any element which is common in both taken only once.

∴ A ∪ B = { 1, 2, 3, 4, 5, 6, 7 }

2) If P = { 1, 2, 3, 4 } and Q = { x | x ∉ N, 1 < x < 8 } then find P ∪ Q.

P = { 1, 2, 3, 4 }

Q = { x | x ∉ N, 1 < x < 8 }

Q = { 2, 3, 4, 5, 6, 7 }

∴ P ∪ Q = { 1, 2, 3, 4, 5, 6, 7 }.

3) If A = { x | x is a multiple of 2 } and

B = { x | x is an odd natural number }. Find A ∪ B.

A = { x | x is a multiple of 2 }

A = { 2, 4, 6, 8, 10, … }

B = { x | x is an odd natural number }.

B = { 1, 3, 5, 7, 9,11, … }

∴ A ∪ B = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11, … }

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**Properties on union of the sets**1) A ∪ A = A

2) A ∪ φ = A

3) A ∪ B = B ∪ A (Commutative property for union)

4) ( A ∪ B ) ∪ C = A ∪ ( B ∪ C ) ( Associative property )

5) U ∪ A = U (Law of U)

**Examples :**1) If A = { 1, 2, 3, 4} and B = { 2, 3, 5, 6, 7 } then find A ∪ B.

**Solution :**A = { 1, 2, 3, 4} and B = { 2, 3, 5, 6, 7 }

A union B is obtained by combining the two sets but if there is any element which is common in both taken only once.

∴ A ∪ B = { 1, 2, 3, 4, 5, 6, 7 }

2) If P = { 1, 2, 3, 4 } and Q = { x | x ∉ N, 1 < x < 8 } then find P ∪ Q.

**Solution :**P = { 1, 2, 3, 4 }

Q = { x | x ∉ N, 1 < x < 8 }

Q = { 2, 3, 4, 5, 6, 7 }

∴ P ∪ Q = { 1, 2, 3, 4, 5, 6, 7 }.

3) If A = { x | x is a multiple of 2 } and

B = { x | x is an odd natural number }. Find A ∪ B.

**Solution :**A = { x | x is a multiple of 2 }

A = { 2, 4, 6, 8, 10, … }

B = { x | x is an odd natural number }.

B = { 1, 3, 5, 7, 9,11, … }

∴ A ∪ B = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11, … }

**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