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}

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The blue region is A ∪ B

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Properties of Union
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- 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)

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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
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)
**

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
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= A
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∪ (B ∪ C) (
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Associative law )
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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)

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

###
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A ∪ A
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= A (Idempotent law
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**
)
**

A U A = {1, 2, 3, 4} ∪ {1, 2, 3, 4}

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

###
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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