Union of sets, Properties of union

Union of Sets Definition in Set Theory

In set theory and discrete mathematics, a set is defined as a collection of distinct elements grouped together as a single unit. These elements may be numbers, letters, objects, or even people. Sets help us organize and analyze data based on common properties, which makes them extremely useful in mathematics, probability, logic, and computer science.

The union of sets is one of the most basic and important set operations. It represents the process of combining all elements from two or more sets into one set, without repeating any element.

If $A$ and $B$ are two sets, the union of $A$ and $B$ includes every distinct element that belongs to either set $A$, set $B$, or both.

Notation of the Union of Sets

The standard symbol used to represent the union operation is $\cup$

This symbol is written between the sets, which is called infix notation.

Symbolically, $A \cup B = \{x : x \in A \text{ or } x \in B\}$

This means any element that appears in $A$, in $B$, or in both will be included in the resulting union set.

How Union of Sets Works (Concept)

You can think of union as combining or merging groups.
Nothing is removed, and duplicates are not counted twice.
All unique elements are simply collected into one larger set.

Union of Sets Examples with Solutions

Example 1: Numerical Sets

Let $A = \{1,3,5,7\}$
$B = \{2,4,6,8\}$

Then, $A \cup B = \{1,2,3,4,5,6,7,8\}$

Here, all elements from both sets are combined into one set.

Example 2: Alphabet Sets

Let $A = \{d, e, g, y, c\}$
$B = \{a, s, d, f\}$ Then, $A \cup B = \{a, c, d, e, f, g, s, y\}$

Notice that $d$ appears in both sets but is written only once, because sets do not allow duplication.

Union of Sets Venn Diagram

Let $A$ and $B$ be any two sets. The union of sets $A$ and $B$ is the set that combines the elements in both sets without duplication. The union of sets is denoted by '$\cup$ '.

Symbolically, we write $\mathrm{A} \cup \mathrm{B}=\{\mathrm{x}: \mathrm{x} \in \mathrm{A}$ or $\mathrm{x} \in \mathrm{B}\}$.

The Venn diagram of the union of sets is

A Union B Formula

The union of sets A and B refers to a set that includes all elements that are in A, in B, or in both. It is written as $A \cup B$ and is read as ”A union B” or ”A or B.” This formula is used to find the combined elements of sets A and B, without repeating any element.

Mathematically, the union of A and B is defined as: $A \cup B=\{x: x \in A$ or $x \in B\}$.

Formula for Number of Elements in A union B

Consider two sets, A and B. To determine the cardinal number of the union of these sets, we say

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$

Here,

$n(A \cup B)$ = Total number of elements in $A \cup B$.

$n(A)$ = Number of elements in $A$.

$n(B)$ = Number of elements in $B$.

$n(A \cap B)$ = The number of elements that are common to both $A$ and $B$ also called the cardinality of set $A \cap B$, i.e. $A$ intersection $B$

Properties of the Union of Sets

Below are the properties of union of sets discussed in detail:

1. Commutative Property: This signifies that the union of sets is independent of interchangeable properties.

$A \cup B=B \cup A$

2. Associative Property: This signifies that the union of three sets can be interchangeable.

$(A \cup B) \cup C=A \cup(B \cup C)$

3. Law of identity element: When we take the union of a finite set from a null set, the original set comes ( $\phi$ is the identity of Null Set).

$\mathrm{A} \cup \phi=\mathrm{A}$

4. Idempotent Property: This signifies that the union of the same set is itself.

$A \cup A=A$

5. Property of universal set: When we take union from a universal set then a universal set will come.

$U \cup A=U$

Note: If $A$ is a subset of $B$, then $A \cup B=B$

6. The union of any two sets results in a completely new set that contains the elements present in both the initial sets.

7. The resultant set contains all elements present in the first set, the second set, or elements in both sets.

8. The union of two disjoint sets results in a set that includes elements of both sets.

9. According to the commutative property of the union, the order of the sets considered does not affect the resultant set.

10. Cardinality of the Union of two sets is always less than or equal to the sum of the cardinalities of the two sets themselves.

$n(A \cup B)$ $<=n(A)+n(B)$

How to Find the Union of Sets Step-by-Step

Finding the union of sets in set theory means combining all distinct elements from two or more sets into one single set. The resulting set contains every element that appears in either set, without duplication.

This method is widely used in discrete mathematics, Venn diagrams, probability, and logical reasoning problems.

Follow these simple steps.

Step 1: Write the Given Sets Clearly

Start by listing all elements of each set properly using roster form.

Example: $A = \{1, 2, 3\}$
$B = \{3, 4, 5\}$

Writing them clearly helps avoid missing or repeating elements.

Step 2: Combine All Elements from Both Sets

Now collect every element from both sets into one list.

From $A$ → $1, 2, 3$
From $B$ → $3, 4, 5$

Combined list → $1, 2, 3, 3, 4, 5$

Step 3: Remove Duplicate Elements

Remember, sets never allow repetition.
If an element appears more than once, keep it only once.

After removing duplicates:

$1, 2, 3, 4, 5$

Step 4: Write the Final Union Set

Now write the result inside curly brackets.

$A \cup B = \{1, 2, 3, 4, 5\}$

This is the union of sets.

Quick Formula for Union of Sets

Mathematically,

$A \cup B = \{x : x \in A \text{ or } x \in B\}$

This means any element belonging to $A$, $B$, or both is included.

Example 2: Alphabet Sets

Let

$A = {a, b, c}$
$B = {b, d, e}$

Combine → $a, b, c, b, d, e$
Remove duplicates → $a, b, c, d, e$

Final answer:

$A \cup B = {a, b, c, d, e}$

Difference Between Union and Intersection of Sets

In set theory and discrete mathematics, union and intersection are two fundamental set operations used to combine or compare sets. The union of sets collects all elements, while the intersection of sets keeps only the common elements. Understanding this difference is essential for Venn diagrams, probability problems, and logical reasoning questions.

Here’s a clear comparison table for quick revision:

Quick Memory Trick

If it helps, think like this:

• Union → “add everything”

• Intersection → “keep only common”

Solved Examples based on Union of Sets

Example 1: If set $\mathrm{A}=\{1,3,5\}$ and $\mathrm{B}=\{3,5,7\}$. Also $P=A \cup B$ and $Q=B \cup A$. Then which of the following is true?
1) $P=Q$
2) $P \subset Q$
3) $P \neq Q$
4) $Q \subset P$

Solution
In this Question,
As $A \cup B=B \cup A$
So, $P=Q$.
Hence, the answer is option 1.

Example 2: If $A=\{5,6,7\}$ and $B=\{\}$, then the value of $A \cup B$ is
1) \{\}
2) $\phi$
3) $\{5,6,7\}$
4) $\{5\}$

Solution
$\mathrm{A} \cup \phi=\mathrm{A}$
Using this property, the union in this question will be $\{5,6,7\}$.
Hence, the answer is the option 3.

Example 3 : What is the union of set A = {1,2,3,4} with a set $B=\left\{x: x^2-5 x+6=0\right.$ or $\left.x^2-5 x+4=0\right\}$?

1) $U$
2) $B$
3) None of these
4) Both (1) and (2)

Solution

Let A and B be any two sets. The union of A and B is the set which consists of all the elements of A and all the elements of B, the common elements being taken only once.

And $A ∪ A = A$

In this Question,

$\begin{aligned} & B=\left\{x: x^2-5 x+6=0 \text { or } x^2-5 x+4=0\right\} \\ & x^2-5 x+6=0 \Rightarrow x=2,3 \\ & x^2-5 x+4=0 \Rightarrow x=1,4 \\ & \Rightarrow B=\{1,2,3,4\}\end{aligned}$

Thus $A = B$

Thus $A \cup B=A=B$

Hence, the answer is option 3.

Example 4: If $\mathrm{n}(\mathrm{U})=20, \mathrm{n}(\mathrm{A})=12, \mathrm{n}(\mathrm{B})=9, n(A \cap B)=4$ where U is the universal set, then $n(A \cup B)^c=$

Solution

$ \begin{aligned} & n(A \cup B)=n(A)+n(B)-n(A \cap B)=12+9-4=17 \\ & n(A \cup B)^{\prime}=n(U)-n(A \cup B)=20-17=3 \end{aligned} $
Hence, the answer is 3.

Example 5: Find the union of the sets $\{3,7,9\},\{5,8,1\}$ and $\{2,8,3\}$.
1) $\{1,2,3,4,5,8,7,9\}$
2) $\{1,2,3,5,7,8,9\}$
3) $\{1,2,3,4,7,8,9\}$
4) $\{1,2,5,7,8,9\}$

Solution
The union of these three sets is the set containing all the elements in these three sets. Option (2) is correct.

Options (1) and (3) have 4, which is not present in any of the given sets.
Option (4) is missing 3.
Hence, the answer is option 2.

List of Topics Related to the Union of Sets

Understanding the union of sets becomes easier when you're familiar with the foundational set theory topics that support it. This section lists essential concepts like roster and set-builder form, intersection, complement, power set, and more that are directly linked to mastering union operations in mathematics.

NCERT Resources

Accessing the right NCERT resources for Class 11 Sets can significantly improve your understanding of the topic. This section provides high-quality study materials including NCERT solutions, revision notes, and exemplar problems to support effective exam preparation.

NCERT Solutions for Class 11 Chapter 1 Sets

NCERT Notes for Class 11 Chapter 1 Sets

NCERT Exemplar for Class 11 Chapter 1 Sets

Practice Questions on Union of Sets

Mastering any mathematical concept comes through continuous practice. To help strengthen your understanding of the topic, we have given below some practice questions on Union of sets. They will test your knowledge of formulas, important properties and general application of knowledge.

Union of Set - Practice questions

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