Intersection of Set, Properties of Intersection

Intersection of sets is an important concept in sets, working on the similarity between two or more sets. Consider two groups of boys. The boys from one group is interested in outdoor games and the other boys are interested in indoor games. Intersection of sets deals with the group of boys interested in both outdoor and indoor games. This concept of intersection of sets is used to highlight the commonalities of the group.

This Story also Contains

1. Intersection of sets
2. Properties of intersection of Sets
3. Intersection Vs Union of Sets
4. Real-Life Examples of Set Intersection
5. Solved Examples Based On the Intersection of Sets

In this article, we will cover the concept of the intersection of sets and its properties. 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 five questions have been asked on this concept, including one in 2021, and five in 2023.

Intersection 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. Intersection of sets represent the common elements in the sets considered.

Intersection of Sets Definition

The intersection of sets $A$ and $B$ is the set of all elements which are common to both $A$ and $B$. The symbol ' $\cap$ ' is used to denote the intersection of sets.

Symbolically, we write $A \cap B=\{x: x \in A$ and $x \in B\}$

If $A$ and $B$ are two sets such that $A \cap B=\varphi$, then $A$ and $B$ are called disjoint sets.

Intersection of set symbol

The symbol for the intersection of sets is " $\cap$ ". For any two sets $A$ and $B$, the intersection, $A \cap$ $B$ (read as $A$ intersection B) lists all the elements present in both sets (common elements of $A$ and B).

How to Find Intersection of Sets?

To find the intersection of the set, we can use the following steps:

Step 1: Compare the elements of the given sets.

Step 2: Select the common elements between both sets.

Step 3: Add the selected elements in the resultant set.

Step 4: Repeat above steps for all the given sets.

Step 5: The resultant set obtained represents intersection of sets.

Now let us look into an example of set intersection for better understanding.

Example of Set Intersection

1. Let $A=\{i,a,f,h,s\}$ and $B=\{f,m,s,h,a,i\}$, then $A \cap B=\{a,s,h,i,f\}$

2. Let $A=\{2,4,6,8\}$ and $B=\{2,3,5,8\}$, then $A \cap B=\{2,8\}$

3. Let $A=\{2,4,6,8\}$ and $B=\{1,3,5,7\}$. Then $A$ and $B$ are disjoint sets because there are no elements that are common to A and B.

4. let $A=\{2,4,6,8\}, B=\{2,3,8\}$, and $C=\{2,3,5,8\}$, then $A \cap B \cap C=\{2,8\}$

Cardinality of Intersection of Sets Formula

The cardinal number of a set is the total number of elements present in the set. For example, if $\operatorname{Set} A=\{1,2,3,4,5,7,8\}$, then the cardinal number (represented as $\mathrm{n}(\mathrm{A})$ ) $=8$.

Consider two sets $A$ and $B$. Let $A=\{2,4,5,9,10,11,18,21\}, B=\{1,2,3,5,7,8,11,12,13\}$ and $A \cap B=\{2,5,11\}$, and the cardinal number of $A$ intersection $B$ is represented by $n(A \cap B)=3$.

The cardinality of $A \cap B$ can also be found by the formula, $n(A \cap B)=n(A)+$ $n(B)-n(A \cup B)$. Let's verify this formula for the above example, where $n(A)=8, n(B)=9$, and $A \cup B=\{1,2,3,4,5,6,7,8,10,11,12,13,14,21\}$. Note that $n(A \cup B)=14$ here. Then

$
\begin{aligned}
& n(A \cap B)=n(A)+n(B)-n(A \cup B) \\
& n(A \cap B)=8+9-14 \\
& n(A \cap B)=3
\end{aligned}
$

Venn Diagram of Intersection of Sets

Now let us look into the venn diagram of intersection of sets.

Intersection of two sets

The intersection of two sets signifies that the common element is present in both sets.

Intersection of three sets

The intersection of three sets signifies that the common element is present in all three sets.

Properties of intersection of Sets

The properties of intersection of sets include commutative law, associative law, law of $\phi$ and $U$, idempotent law and distributive law.

Commutative law: This law states that the intersection of two sets is interchangeable which means from left or right both are the same.
$A \cap B=B \cap A$.
Associative law: This law states that the intersection of three sets is associative which means the intersection of any two sets with third sets are the same.
$(A \cap B) \cap C=A \cap(B \cap C)$.
Law of $\phi$ and $U$: This law states that the intersection of the set to the universal set results in the same set. $\mathrm{A} \cap \mathrm{U}=\mathrm{A}$
Idempotent law: This law states that the intersection of the set to itself results in the same set.
$A \cap A=A \mid$

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

Distributive law: This law states that,

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

Intersection Vs Union of Sets

Real-Life Examples of Set Intersection

Some real-life examples of intersection of sets are

• Consider two gardens $A$ and $B$. The first garden has the set of flowers $\{rose, tulip, sunflower, jasmine\}$. The second garden has the set of flowers $\{ lily, lotus, rose, tulip, daisy\}$. The intersection of sets of flowers in the gardens $A$ and $B$ is $\{rose, tulip\}$.
• Let us consider two genres of books in library. Let $A$ be the set of books of science fiction genre and $B$ be the set of books of mystery genre. The intersection of $A$ and $B$ is the set of science fictional mystery books.
• Let $A$ be the set of all students who play football and $B$ be the set of all students who play cricket. Then the intersection of sets $A$ and $B$ is the set of all students who play both football and cricket.

Recommended Video Based on the Intersection of Sets

Solved Examples Based On the Intersection of Sets

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

Solution:
As we learned
According to the property,
$A \cap B=B \cap A$
$\mathrm{Q}=\mathrm{R}$.
Hence, the answer is the option 2.

Example 2: Which of the following is the associative property of intersection?
1) $(A \cup B) \cup C=A \cup(B \cup C)$
2) $(A \cap B) \cup C=A \cup(B \cap C)$
3) $A \cap(B \cup C)=(A \cap B) \cup C$
4) $(A \cap B) \cap C=A \cap(B \cap C)$

Solution:
As we learned
The Associative Property of an intersection is:
$A \cap(B \cap C)=(A \cap B) \cap C$
Hence, the answer is the option 4.

Example 3: If $A=\phi$ and $A \cap B=C$, then which of the following is true?
1) $A=B$
2) $A=C$
3) $B=C$
4) $\mathrm{B}=\phi$

Solution:
As we learned that $\varphi \cap \mathrm{A}=\varphi$
So, $C=A \cap B=\phi \cap B=\phi$
Thus $\mathrm{A}=\mathrm{C}$.
Hence, the answer is the option 2.

Example 4: If $A$ and $B$ are equal sets, then which of the following is NOT true?
1) $A \cap B=A$
2) $A \cap B=B$
3) $A \cup B=A \cap B$
4) $A \cap B=\phi$

Solution:
$A \cap B=A \cap A=A$ : so option (1) is true
As $A=B$, so $A \cap B=A=B$ : so option (2) is true
Also, $A \cup B=A \cup A=A$
As $A \cap B=A$, so $A \cup B=A \cap B$ : so option (3) is true.
Therefore, the incorrect option is 4 .
Hence, the answer is the option 4.

Example 5: Given $A \cap B=\{5,7,9\}, A \cap C=\{3,8,7\}$. Find the value of $A \cap(B \cup C)$
1) $\{7\}$
2) $\{3,8,7\}$
3) $\{3,5,7,8,9\}$
4) $\{5,7,9\}$

Solution:

$
\begin{aligned}
& A \cap(B \cup C)=(A \cap B) \cup(A \cap C) \quad \text { [Using Distributive Property] } \\
& =\{5,7,9\} \cup\{3,8,7\} \\
& =\{3,5,7,8,9\}
\end{aligned}
$

Hence, the answer is the option 3.

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