A and B are the two sets that contain elements. The union of two sets A and B is the combination of the elements in A and B sets. Union of the sets A and B is denoted as A U B. If there are any common elements in two sets, we need to write only once in the union of two sets. The symbol of the Union is denoted as ‘U’ and it is called a universal set. For example,

A = {1, 4, 5, 3} and B = {2, 7, 8, 9}

A U B = {1, 2, 3, 4, 5, 7, 8, 9}.

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## Union of Sets Venn Diagram Representation

Let us assume a Universal Set U where A, B are Subsets of the Universal Set. Union of Sets is defined as the set of all the elements that exist in Set A and Set B or both the elements in Set A, Set B together. Union of Sets is denoted by the Symbol ‘U’. Venn Diagram Representation of Union of Sets is given below.

A U B = {x: x ∈ A or x ∈ B}.

### Properties of the Union of Sets

There are different Properties of Union of Sets and we have explained all of them in detail by considering few examples. They are as such

(i) Commutative Law

(ii) Associative Law

(iii) Identity Law

(iv) Idempotent Law

(v) Domination Law

**(i) Commutative Law**

The commutative law is, A ∪ B = B ∪ A.

If A = {1, 2, 3} and B = {3, 4, 5} then

A ∪ B = {1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5} ——(1).

B ∪ A = {3, 4, 5} ∪ { 1, 2, 3} = {3, 4, 5, 1, 2} ——(2).

(1) = (2)

The elements of the union set A ∪ B and B ∪ A are the same. So, the commutative law is satisfied.

**(ii) Associative Law**

The Associative law contains three sets A, B, and C and it is (A ∪ B) ∪ C = A ∪ (B ∪ C).

For example, A = {1, 2}, B = {3, 4, 5} and C = {1, 4, 6}.

Then, A ∪ B = {1, 2} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}.

(A ∪ B) ∪ C = {1, 2, 3, 4, 5} ∪ {1, 4, 6} = {1, 2, 3, 4, 5, 6}—–(1).

(B ∪ C) = {3, 4, 5} ∪ { 1, 4, 6} = { 1, 3, 4, 5, 6}.

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

(1) = (2).

The elements of the union sets (A ∪ B) ∪ C and A ∪ (B ∪ C) are the same. So, an Associative law is satisfied.

**(iii) Identity Law**

The identity law is A ∪ Ø = A.

Ø is an empty set. That is Ø = { }and A = {1, 2, 3}.

A ∪ Ø = {1, 2, 3} ∪ { } = {1, 2, 3}.

A ∪ Ø = A.

So, the Identity law is satisfied.

**(iv) Idempotent Law**

An idempotent law is A ∪ A = A.

If A = {2, 6, 8, 9}, then

A ∪ A = {2, 6, 8, 9} ∪ {2, 6, 8, 9} = {2, 6, 8, 9}.

A ∪ A = A.

So, an Idempotent law is satisfied.

**(v) Domination Law**

The domination law is A ∪ U = U.

If A = {a, b, c, d} and U = {a, b, c, d, e, f, g, h, i}.

A ∪ U = {a, b, c, d, e, f, g, h, i}.

Then, A ∪ U = U.

So, the Domination law is satisfied.

### Solved Examples of Union of Sets

**Problem 1.**

’∪’ be a universal set and A and B are subsets of ∪. If A = {2, 5, 1, 3} and B = {9, 8, 7, 6, 5}, then find A ∪ B?

**Solution:**

As per the given details A and B are the subsets of ∪.

A = {2, 5, 1, 3} and B = {9, 8, 7, 6, 5}.

A ∪ B = {2, 5, 1, 3} ∪ {9, 8, 7, 6, 5}.

A ∪ B = {1, 2, 3, 5, 6, 7, 8, 9}.

**Problem 2.**

If A = {a, b, c, d, e} and B = {Ø} then find the union of A and B?

**Solution:**

As per the given information A = {a, b, c, d, e} and B = {Ø}.

A ∪ B = {a, b, c, d, e} ∪ {Ø}

A ∪ B = {a, b, c, d, e} = A.

Therefore, A ∪ B is equal to A.

**Problem 3.**

Three sets are there A = {a, b}, B = { b, c, d} and C = { c, d, f}. Check whether the union of sets are satisfies the associative law or not?

**Solution:**

As per the given details, the three sets are A = {a, b}, B = {b, c, d} and C = {c, d, f}.

We need to check whether the above sets are satisfying the associative law or not.

An Associative law is (A ∪ B) ∪ C = A ∪ (B ∪ C).

A ∪ B = {a, b} ∪ {b, c, d}

A ∪ B = {a, b, c, d}.

(A ∪ B) U C = {a, b, c, d} U {c, d, f}

(A U B) ∪ C = {a, b, c, d, f} —–(1).

B ∪ C = {b, c, d} ∪ {c, d, f}

B ∪ C = {b, c, d, f}.

A ∪ (B ∪ C) = {a, b} ∪ {b, c, d, f} = {a, b, c, d, f} —–(2).

(1) = (2).

Therefore, the given sets satisfy the associative law.

**Problem 4.**

If U = {1, 5, 6 8, 9, 4, 3, 2, 7} and A = {2, 4, 5}, then find the union of A and U?

**Solution:**

As per the given information

U = {1, 5, 6 8, 9, 4, 3, 2, 7} and A = {2, 4, 5}.

The union of A and U is

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

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

Therefore, A ∪ U = U, and this is also called domination law.