Union of Sets

Union of sets is one of the set operations that is used in set theory. In addition to the union of sets, the other set operations are difference and intersection. All the set operations are represented by using a unique operator. The union of sets is analogous to arithmetic addition. The union of two given sets is the set that contains all the elements present in both sets. The symbol for the union of sets is "∪''. For any two sets A and B, the union, A ∪ B (read as A union B) lists all the elements of set A as well as set B. Thus, for two given sets, Set A = {1,2,3,4,5} and Set B = {3,4,6,8}, A ∪ B = {1,2,3,4,5,6,8}

In this article, you will learn about the union of sets, its definition, properties with solved examples.

The union of any two or more sets results in a completely new set that contains a combination of elements that are present in both those two or more given sets. The union of sets is represented by using the word ‘or’. Let's consider two sets A and B. The union of A and B will contain all the elements that are present in A or B or both sets. The set notation used to represent the union of sets is ∪. The set operation, that is the union of sets, is represented as:

A ∪ B = {x: x ∈ A or x ∈ B}. Here, x is the element present in both sets, A and B.

Finding a Union of Sets

Let's look at the following example to understand the process of finding the union of sets. We have two sets A and B as A = {a, b, j, k} and B = {h, t, k, c}. We need to find out the elements present in the union of A and B.

As per the definition of the union of two sets, the resultant set will include elements that are present in A, in B or both sets. Thus, the elements of both sets are a, b, c, j, k, h, t but since the element k is present in both sets so it will be considered only once as it is common to both the given sets. Therefore, the elements present in the union of sets A and B are a, b, c, j, k, h, t

Notation of Union of Sets

We use a unique mathematical notation to represent each set operation. The mathematical notation that is used to represent the union between two sets is '∪'. This operator is called infix notation and it is surrounded by the operands.

Consider two sets P and Q, where P = {2,5,7,8} and Q = {1,4,5,7,9}. P ∪ Q = {1,2,4,5,7,8,9}.

Venn Diagrams refer to the diagrams that are used to represent or explain the relationship between the given set operations. Any set operation can be represented by using a Venn diagram. Venn diagrams represent each set using circles. Let's see how to use the Venn diagram to represent the union of two sets. For this, we first need a universal set, of which the two given sets P and Q are the subsets. The following Venn diagram represents the union between the sets P and Q.

In the above-given Venn diagram, the blue-colored region shows the union of sets P and Q. This further represents that the union between these sets includes all the elements that are present in P or Q or both sets. Although the union operation between two sets has been used here, the Venn diagram is often used to represent the union between multiple sets, provided that the sets are finite.

In this section, you will be learning about some of the important properties of the union of sets. It is essential to take these properties into consideration while performing a union of sets.

Commutative Property

As per the commutative property of the union, the order of the operating sets will not affect the resultant set. This means that if the position of the operands is changed, the solution will stay the same and it will not be affected. In mathematical terms, we can say that: A ∪ B = B ∪ A

Let's consider two sets P and Q:

P = {a, m, h, k, j}, Q = {2, 3, 4, 6}

To prove that the commutative property holds for these sets, we first need to solve the left-hand side of the equation, which is:

P ∪ Q = {a, m, h, k, j} U {2, 3, 4, 6} = {a, m, h, k, j, 2, 3, 4, 6}

Now, we will be solving the right-hand side of the equation:

Q ∪ P = {2, 3, 4, 6} U {a, m, h, k, j} = {a, m, h, k, j, 2, 3, 4, 6}

Now, we can conclude that the commutative property is true for the union of given sets.

Associative Property

As per the associative property of union, when the sets are grouped using parentheses, the result will not be affected. This means that when the parentheses’ position is changed in any expression of sets that involves union, then the resultant set will not be affected by this. In mathematical terms,

(A ∪ B) ∪ C = A ∪ (B ∪ C), where A, B, and C are any finite sets.

Let's prove that the associative property of union holds true for the following sets:

A = {2, 3, 4}, B = {2, 5, 6}, C = {1, 6, 9}

Let's solve the left side of the above equation:

(A ∪ B) = {2, 3, 4} U {2, 5, 6} = {2, 3, 4, 5, 6}

(A ∪ B) ∪ C = {2, 3, 4, 5, 6} U {1, 6, 9} = {1, 2, 3, 4, 5, 6, 9}

Now, let's solve the right side of the equation:

(B ∪ C) = {2, 5, 6} ∪ {1, 6, 9} = {1, 2, 5, 6, 9}

A ∪ (B ∪ C) = {2, 3, 4} ∪ {1, 2, 5, 6, 9} = {1, 2, 3, 4, 5, 6, 9}

From the left and right sides of the equations, we can conclude that the associative property of union holds true for the given sets A, B, and C.

Idempotent Property

The idempotent property states that the union of any set with the same set will result in the set itself. It can be shown mathematically as A ∪ A = A.

Let's prove this for A = {2,4,6,8,10}

Thus, A ∪ A = {2,4,6,8,10} ∪ {2,4,6,8,10} = {2,4,6,8,10} = A

Property of Ⲫ/ Identity Law

As per the property of a null set, the union of any set with a null set or an empty set will result in the set itself. Mathematically, we can write it as A ∪ Ⲫ = A.

Let's prove this for A = {p,q,r}

Thus, A∪∅ = {p,q,r} ∪ {} = {p,q,r}

Property of Universal Set

As per the property of the universal set, the union of the universal set with any set results in the universal set. Mathematically it can be represented as A ∪ U = U.

Let's prove this for A = {a,e} and U = {a,b,c,d,e,f,g,h}

then A∪U = {a,e} ∪ {a,b,c,d,e,f,g,h} = {a,b,c,d,e,f,g,h} = U

Related Articles on Union of Sets

Check out the following pages related to the union of sets

• Finite and Infinite Sets
• Sets Formula

Important Notes on Union of Sets

Here is a list of a few important points related to the union of sets.

• Union of any two sets results in a completely new set that contains the elements that are present in both the initial sets.
• The resultant set contains all elements that are present in the first set, the second set, or elements that are in both sets.
• The union of two disjoint sets results in a set that includes elements of both sets.
• As per the commutative property of the union, the order of the operating sets does not affect the resultant set.
• To determine the cardinal number of the union of sets, use the formula: n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

What is the Union of Sets in Math?

In math, the union of any two sets is a completely new set that contains elements that are present in both the initial sets. The resultant set is the combination of all elements that are present in the first set, the second set, or elements that are in both sets. For example, the union of sets A = {0,1,2,3,4} and B = {13} can be given as A ∪ B = {0,1,2,3,4,13}.

What is the Difference Between Intersection and Union of Sets?

Union of any two sets results in a completely new set that contains elements that are present in the first set, the second set, or elements that are in both sets. Whereas, the intersection of sets will contain elements that are common in both sets. Consider two sets A = {1,2} and B = {2,3}. Here, the union of A and B will be A ∪ B = {1,2,3} whereas the intersection of A and B will be A ∩ B = {2}.

What is the Symbol For Union of Sets?

The mathematical notation that is used to represent the union of sets is '∪'. This operator is called infix notation and is surrounded by the operands.

What is the Commutative Property of the Union of Sets?

As per the commutative property of the union, the order of the operating sets does not affect the resultant set. On changing the position of the operands, the solution will stay the same, it will not be affected. In mathematical terms, we can say that: A ∪ B = B ∪ A.

What is the Associative Property of the Union of Sets?

As per the associative property of union, when the sets are grouped using parentheses, then the resultant set does not get affected when the parentheses’ position is changed in any expression of sets that involves union. In mathematical terms, (A ∪ B) ∪ C = A ∪ (B ∪ C), where A, B, and C are any finite sets.

What is the Idempotent Property of the Union of Sets?

The idempotent property states that the union of any set with the same set will result in the set itself. It can be shown mathematically as A ∪ A = A.

What is the Property of Ⲫ in Union of Sets?

As per the property of a null set, the union of any set with a null set or an empty set will result in the set itself. Mathematically, we can write it as A ∪ Ⲫ = A.

What is the Union of Sets a and b?

The union of two sets A and B is defined as the set of all the elements which are present in set A and set B or both the elements in A and B altogether. The union of the sets a and is denoted as 'a ∪ b'.

What is the Process of Finding a Union?

Union of two sets can be considered as the least set comprising the elements of both sets. For finding the union of two sets, follow the steps given below:

• Step 1: Consider the two or more given sets.
• Step 2: Pick up the elements of two or more given sets and prepare a resultant set in which no element is repeated.
• Step 3: Represent the union of sets using the symbol '∪'.

For example, the union of X = {11,12,13,14,15,16,17,18,19,20} and Y = {13,17,21} = X∪Y = {11,12,13,14,15,16,17,18,19,20,21}.

What is the Cardinality of the Union of Sets A and B?

For the finite sets A and B, the number of elements is counted using one-to-one correspondence, but duplicates are not into account. For example, if the union of sets = {3, 2, 1, 2, 3}, then it has cardinality 3.