Union of Sets | Formula, Examples & Venn Diagrams for Union of Sets - GeeksforGeeks (2025)

Union of two sets means finding a set containing all the values in both sets. It is denoted using the symbol '∪' and is read as the union.

Example 1:
If A = {1, 3. 5. 7} and B = {1, 2, 3} then A∪B is read as A union B and its value is,
A∪B = {1, 2, 3, 5, 7}

Example 2:
If A = {1, 3. 5.7} and B = {2, 4} then A∪B is read as A union B and its value is,
A∪B = {1, 2, 3, 5, 7}

Example 3:
If A = {1, 2, 3}, B = {2, 4} and C = {1, 3, 4}
A∪B∪C = {1, 2, 3, 5}

The union of the set can also be represented using the Venn Diagrams. For example, if we have set A and set B which have some values in common then their Venn diagram is represented in the image below,

We can represent any set using the Venn diagram in the Venn diagram explained above the rectangle represents the Universal set, and set A and set B are represented using the circles. The common area of the two sets represents the intersection of the two sets and both the circles combined along with the common area represent the union of the set.

Mathematical Definition

The union of any two or more sets is a set that contains all the elements of the previous sets. The union of two sets is equivalent to the logical operation OR and it means any of the given values, for example, if we take a set A = {a, e, i, o, u} and set B = {a, b, c, d, e} then OR operations signifies any of the value of set A and set B and this can be written as A∪B and its value is equal to,

A∪ B = {a, b, c, d, e, i, o, u}

Here, the union set contains all the values either in set A or set B.

In general, for two sets, set A and set B we represent the union of sets in set builder form as,

A ∪ B = {x: x ∈ A or x ∈ B}

Also Read:

• Complement Of Set
• Set Theory

How to Find Union of Sets?

We can easily find the union of two sets by taking all the elements of both sets and removing the common elements. Let's learn this concept through an example.

Example: Find the union of the sets, set A = {p, q, r, s, t, u} and set B = {s, t, u, v, w,}.

Solution:

The union of set A and set B is found by taking all the elements of set A and set B and taking the common element only once.

A∪ B = {p, q, r, s, t, u, v, w}

Here, all the elements of set A and set B are taken and the elements which appear twice (s,t,u) are taken only once.

Union of Sets Symbol

Union of the sets is represented using the symbol "∪". It is placed between two sets whose union is to be found. We read this symbol as "union". Example A∪B is read as A union B, furthermore we can also find the union of two or more sets as the union of set A, set B, and set C is represented as, A∪B∪C and is read as A union B union C.

Note: We can find the union for any number of finite or countable infinite sets.

Union of Sets Formula - A U B

As we already discussed set A union B contains all the elements of set A as well as set B, but there are some formulas related to the A U B operation that helps us calculate many things. One such formula involving union of two sets, is discussed as follows:

Formula for Number of Elements in A union B

To find the number of elements in the set of A union B, we can use the following formula:

n(A U B) = n(A) + n(B) – n(A ∩ B)

Where,

• n(A U B) is the number elements in A U B,
• n(A) is the number of elements in A,
• n(B) is the number of elements in B, and
• n(A ∩ B) is the number of elements that are common to both A and B.

Note: n(A) or |A| is called the cardinality of the set A i.e., the number of elements set A contains.

Also Check: Intersection of Sets

Properties of Union of Sets

The intersection of set has various properties. The table below discusses the properties of the union of the set.

Now let's learn about these properties in detail.

Commutative Property

The commutative property of the union of the set explains that the order in which the union of two sets is taken is not important. For example, if take the union of two sets, set A and set B then the value of A ∪ B is equal to the B ∪ A. We can write this property as,

A ∪ B = B ∪ A

Example: Take two sets, set A = {1, 3, 5, 7}, and set B = {a, b, c, d} and find their union.

Solution:

Given sets,

A = {1,3,5,7}
B = {a,b,c,d}

See Also

Problem 32 Let \(A\) and \(B\) be two sets ... [FREE SOLUTION]Word Problems on Sets | Solved Examples on Sets | Problems on Sets| Ideas on Set5.3: Properties of Set OperationsLet a and B Be Two Sets that N ( a ) = 16 , N ( B ) = 14 , N ( a ∪ B ) = 25 Then, N ( a ∩ B ) - Mathematics | Shaalaa.com

Now, for proving the commutative property.

A ∪ B = {1,3,5,7} ∪ {a,b,c,d}
⇒ A ∪ B = {1,3,5,7,a,b,c,d}...(i)

Similarly,

B ∪ A = {a,b,c,d} ∪ {1,3,5,7} = {a,b,c,d,1,3,5,7}

As we know the order of elements is not important in sets so,

B ∪ A = {a, b, c, d, 1, 3, 5, 7}
⇒ B ∪ A = {1, 3, 5, 7, a, b, c, d}...(ii)

Thus from (i) and (ii) we say that

A ∪ B = B ∪ A,

Thus, commutative property for union of sets can be varified.

Associative Property

The associative property of the union of the set explains that the order in which the two sets are grouped for finding the union of two or more sets is not important. For example, if take the union of three finite sets, set A, set B, and set C then,

(A ∪ B) ∪ C = A ∪ (B ∪ C)

Example: Take three sets, set P = {1, 3, 5, 7}, set Q = {a, b, c, d}, and set R = {p, q, r, s}. Verify Associative property.

Solution:

Given sets,

P = {1, 3, 5, 7}
Q = {a, b, c, d}
R = {p, q, r, s}

Now, for proving the associative property.

P ∪ Q = {1,3,5,7} ∪ {a,b,c,d} = {1,3,5,7,a,b,c,d}
⇒ (P ∪ Q) ∪ R = {1, 3, 5, 7, a, b, c, d} ∪ {p, q, r, s}
⇒ (P ∪ Q) ∪ R = {1, 3, 5, 7, a, b, c, d, p, q, r, s}...(i)

Similarly,

Q ∪ R = {a,b,c,d} ∪ {p,q,r,s} = {a,b,c,d,p,q,r,s}
⇒ P ∪ (Q ∪ R) = {1, 3, 5, 7} ∪ {a, b, c, d, p, q, r, s}
⇒ (P ∪ Q) ∪ R= {1, 3, 5, 7, a, b, c, d, p, q, r, s}...(ii)

Thus from (i) and (ii) we say that

(P ∪ Q) ∪ R = P ∪ (Q ∪ R)

Thus, the associative property of the union of the set is verified.

Identity Law (Property of Ⲫ)

The Identity Law of the union of the sets states that the union of any set with an identity element will result in the same set. It can be represented as

A ∪ Ⲫ = A

where Ⲫ is the identity set or null set. This is also called the Property of Ⲫ or the Property of identity set.

Example: If A = {1,2,3,4,5,6} prove A ∪ Ⲫ = A

Solution:

Given,

A ∪ Ⲫ = {1, 2, 3, 4, 5, 6} ∪ { } = {1, 2, 3, 4, 5, 6}

⇒ A ∪ Ⲫ = A

Thus, Identity Law is verified.

Property of Universal Set

Property of the Universal Set of the union of the sets states that the union of any set with the universal set will result in the Universal set. It can be represented as

A ∪ U = U

Note: This property is sometimes referred to as Domination Law.

Example: If A = {1,2,3} and U = {1,2,3,4,5,6,7,8} then prove A ∪ U = U

Solution:

Given,

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

⇒ A ∪ U = {1, 2, 3, 4, 5, 6, 7, 8}

⇒ A ∪ U = U

Thus, Property of Universal set is verified.

Idempotent Property

Idempotent property of the union of the sets states that the union of any set with itself will result in the same set. It can be represented as

A ∪ A = A

Example: If A = {1, 2, 3, 4, 5, 6} then verify the idempotent property.

Solution:

Given,

A ∪ A = {1, 2, 3, 4, 5, 6} ∪ {1, 2, 3, 4, 5, 6}

⇒ A ∪ A = {1,2,3,4,5,6}

⇒ A ∪ A = A

Thus, Idempotent Property is verified.

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Union of Sets Examples

Example 1: Find the Union of the sets,

• A = {1, 2, 3, 4, 5, 6}
• B = {5, 6, 7, 8, 9}

Solution:

Given set,

Set A = {1, 2, 3, 4, 5, 6}
Set B = {5, 6, 7, 8, 9}

Union of sets

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

⇒ A∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9}

Example 2: Find the Union of the sets given below,

• P = {a, e, i, o, u}
• Q = {p, q, r, s, t}
• R = {j, k, l, m, n}

Solution:

Given set,

P = {a, e, i, o, u}
Q = {p, q, r, s, t}
R = {j, k, l, m, n}

Thus, P∪ Q∪ R = {a, e, i, o, u} ∪ {p, q, r, s, t} ∪ {p, q, r, s, t}

⇒ P∪ Q∪ R = {a, e, i, o, u, p, q, r, s, t, j, k, l, m, n}

Example 3: Find the union of sets P and Q, if P = {1, 2, 3, 4, 5} and Q = Ⲫ.

Solution:

Given,

Set P = {1,2,3,4,5}

Set Q = Ⲫ

We know that,

P ∪ Ⲫ = P

⇒ P ∪ Q = {1,2,3,4,5} ∪ Ⲫ

⇒ P ∪ Q = {1,2,3,4,5} = P

Example 4: Find the union of Q = Sets of Rational Nimbers and Q^o = Set of Irrational Numbers

Solution:

We know that,

Set of Rational Numbers, Q = {p/q where p, q ∈ z, q ≠ 0}

Set of Irrational numbers, Q^o = {x where x is not a rational number}

Union of these two sets is Q ∪ Q^o we know that,

Q ∪ Q^o = R {Real Numbers}

Thus, the union of the set of rational numbers and the set of irrational numbers is Real Numbers.

Practice Problems on Union of Sets

Problem 1: Let set A={1,2,3,4,5} and setB={4,5,6,7,8}. Find A ∪B.

Problem 2: Given set C={a,b,c} and set D={c,d,e}, calculate C ∪D.

Problem 3: If set E={2,4,6,8,10} and set F={3,6,9,12}, determine E∪F.

Problem 4: Consider two sets: set G={x,y,z} and set H={w,x,y}. Find G ∪H.

Union of Sets Class 11

The union is used to gather all the distinct elements from the sets being considered, providing a comprehensive collection of elements without any repetition. This concept is essential for students in Class 11 as it lays the groundwork for more advanced topics in mathematics. Understanding the union of sets helps students in comprehending how to combine different datasets and analyze the relationships between them. This foundational concept is not only crucial in set theory but also in various applications across different fields of mathematics and science.

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