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Union of sets A & B has all the elements of set A and set B

It is represented by symbol ∪

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

A ∪ B = {1, 2, 3, 4, 5, 6}

The blue region is A ∪ B

##
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Properties of Union
**

- A ∪ B = B ∪ A (Commutative law)
- (A ∪ B) ∪ C = A ∪ (B ∪ C) (Associative law )
- A ∪ ∅ = A (Law of identity element, ∅ is the identity of ∪)
- A ∪ A = A (Idempotent law)
- U ∪ A = U (Law of U)

Let us discuss these laws

Let us take sets

Let A = {1, 2, 3, 4} , B = {3, 4, 5, 6}, C = {6, 7, 8}

and Universal set = U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

###
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A ∪ B
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= B
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∪ A
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(Commutative law
**
**
)
**

A ∪ B = {1, 2, 3, 4, 5, 6}

B ∪ A = {1, 2, 3, 4, 5, 6}

∴
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A ∪ B = B ∪ A
**

###
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(A ∪ B) ∪ C
**
**
= A
**
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∪ (B ∪ C) (
**
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Associative law )
**

A ∪ B = {1, 2, 3, 4, 5, 6}

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

B ∪ C = {3, 4, 5, 6} ∪ {6, 7, 8}

B ∪ C = {3, 4, 5, 6, 7, 8}

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

∴ (A ∪ B) ∪ C = A ∪ (B ∪ C)

###
**
A
**
**
∪
**
**
∅
**
**
=
**
**
A (Law of identity element,
**
**
∅
**
**
is
**
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the identity of ∪
**
**
)
**

In union, all the elements of set A and empty set (∅) will be there.

Since ∅ has no element, the union will have all the elements of set A only.

That is, union will be A

A U ∅ = {1, 2, 3, 4} ∪ {}

A U ∅ = {1, 2, 3, 4} = A

∴ A ∪ ∅ = A

###
**
A ∪ A
**
**
= A (Idempotent law
**
**
)
**

A U A = {1, 2, 3, 4} ∪ {1, 2, 3, 4}

A U A = {1, 2, 3, 4} = A

###
**
U ∪ A = U (Law of U)
**

Union will have all the elements of Universal set and A

Since Universal set has all the elements, union will be the universal set

U ∪ A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} ∪ {1, 2, 3, 4}

U ∪ A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} = U

∴ U ∪ A = U