Union of sets, Properties of union

Consider two sets, the set of 12th standard students from section A and the set of all 12th standard students from section B from a particular school. By combining these two sets, we get the set of all 12th standard students of that school. This process of combining two or more sets is the union of these sets.

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This Story also Contains

1. Union of Sets
2. Properties of Union of Sets
3. Union of Sets Questions

Union of sets is operation on two or more sets in which the elements in the sets are combined to one single set without duplication. Real life examples of union of sets include the set of all students in a class which is a union of two sets, the set of all male students in the class and the set of all female students in the class; the set of books in a library which is the union of sets of books from different genre; etc.

In this article, we will cover the concept of the union of sets class 11. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of two questions have been asked on this concept, including one in 2014, and one in 2020.

Union of Sets

A set is simply a collection of distinct objects, considered as a whole. These objects, called elements or members of the set, can be anything: numbers, people, letters, etc. Sets are particularly useful in defining and working with groups of objects that share common properties.

It is a well-defined collection of distinct objects and it is usually denoted by capital letters A, B, C, S, U, V.

Union of Sets Definition

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

Union of Sets Examples

1. If $A=\{1,3,5,7\}$ and $B=\{2,4,6,8\}$ then $A \cup B$ is read as $A$ union $B$ and its value is, $A \cup B=\{1,2,3,4,5,6,7,8\}$

2. If If $A=\{d, e, g, y, c\}$ and $B=\{a,s,d,f\}$ then $A \cup B=\{a,c,d,e,f,g,s,y\}$

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 union of sets is

Properties of Union of Sets

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 ( $\varphi$ is the identity of Null Set).
$\mathrm{A} \cup \varphi=\mathrm{A}$

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 $A \cup B=B$

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

• The resultant set contains all elements present in the first set, the second set, or elements in both sets.
• The union of two disjoint sets results in a set that includes elements of both sets.
• According to the commutative property of the union, the order of the sets considered does not affect the resultant set.
• To determine the cardinal number of the union of sets, use the formula: $n(A \cup B)$$
  =n(A)+n(B)-n(A \cap B)
  $

Union of Sets Questions

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

Example 3: If $A \cup B=P$ and $B \cup C=Q$, then which of the following is true?
1) $P \cup C=A \cup Q$
2) $P \cup C=B \cup Q$
3) $P \cup C=C \cup Q$
4) $P \cup B=A \cup Q$

Solution
As we have learnt
$(A \cup B) \cup C=A \cup(B \cup C)$
In this question,
$A \cup B=P$
So, $P \cup C=(A \cup B) \cup C$....(i)
$B \cup C=Q$

So, $A \cup Q=A \cup(B \cup C)$ $\qquad$
From (i) and (ii) and Associative property
$P \cup C=A \cup Q$
Hence, the answer is the option 1.

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 the option 2.

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