Union of Sets

The union of sets means combining all the elements from two or more sets into one set without repeating any element. It is denoted using the symbol '∪'.

For example, if we have set A and set B, which have some values in common, then their Venn diagram is represented as,

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, 2, 3}, B = {2, 4} and C = {1, 3, 4} then A∪B∪C is read as A union B union C and its value is,
A∪B∪C = {1, 2, 3, 5}

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}

Finding 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.

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 Formula

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.

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
• Complement Of Set
• Set Theory

Properties of Union of Sets

The intersection of set has various properties.

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.

Given sets,

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

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.

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

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

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

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.

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.

Solved Examples

Example 1: Find the Union of the sets,

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

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}

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.

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 Numbers and Q^o = Set of Irrational Numbers

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

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.