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Union of Sets

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The representation of similar types of data is called the set. Union of a set is the basic operation on the sets which is used to find all the entries of the given sets. It is one of the operators on the set which is used to solve the set theory problems. Union of two sets means finding a set that contains all the values in both the given sets. It is denoted using the symbol ‘∪’ and is read as the union, i.e. 

If A = {1,3.5.7} and B = {2,4,6,8} then A∪B is read as A union B and its value is,

A∪B = {1,2,3,4,5,6,7,8}

Thus, from the above example, it is clear that a set that contains all the elements of set A and set B is called the union of set A and B.

In this article, we will learn about the union of sets, its definition, its properties, and others.

What is Union of Sets?

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 its 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 it is in set A or in 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,

Finding a 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.

Notation of Union of Sets

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.

A Union B 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. 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:

Venn Diagram of Union of Sets

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,

Venn Diagram of Union of Sets

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.

Properties of Union of Sets

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

Properties of Union

Notation

Commutative Property A∪ B = B ∪ A
Associative Property (A ∪ B) ∪ C = A ∪ (B ∪ C)
Identity Law (Property of Ⲫ) A ∪ ∅ = A
Property of Universal Set A ∪  U = U
Idempotent Property A ∪  A = A

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}

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.

Also Check:

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 Qo = 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, Qo = {x where x is not a rational number}

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

Q ∪ Qo = R {Real Numbers}

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

FAQs on Union of Sets

1. What is the Union of Sets in Maths?

In maths, union of sets is the operation on the set which finds a set such that it has all the elements of all the sets of which the union is taken (duplicity of the elements is not allowed.)

2. What is the Symbol for Union of Sets?

The symbol which is used to represent the union of the set is ‘∪’. If we have to find the union of set A and set B we write A ∪ B and it is read as A union B.

3. What is the difference between Intersection and Union of Sets?

In the union of the set all the unique elements of both sets are taken whereas, in the intersection of the set, only the common elements of the set are taken.

4. What is Union of Two Sets?

The union of the two sets is the set which contains all the elements of set A and set B but the duplicity of the element is not allowed.

5. How to find Union of Sets?

To find the union of the set follow the steps given below.

Step 1: Compare all the elements of the given set

Step 2: List all the elements of the first set.

Step 3: List all those elements of the second set which are not in the first set.

Step 4: Similarly, repeat step 3 for all the given sets.

Step 5: The resultant set so obtained is the union of all the given sets.

6. What is the definition of Union in Sets?

In set theory, the union of two or more sets refers to the combination of all distinct elements present in any of the given sets, resulting in a new set containing those unique elements.

7. What are the Applications of Sets?

Sets have diverse applications:

  • Mathematics: Fundamental for defining relationships.
  • Computer Science: Data structures, databases, and algorithms.
  • Statistics: Probability and data analysis.
  • Social Sciences: Modeling networks and populations.
  • Engineering: Properties and systems.
  • Programming: Managing and manipulating data.
  • Physics and Biology: Modeling complex systems.
  • Finance and Medicine: Risk assessment and data classification.
  • Operations Research: Optimization problems.

8. What does and ∪ mean in Math?

In maths, ∩ means intersection of two sets and ∪ means union of sets.


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Last Updated : 04 Sep, 2023
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