Intersection of Set, Properties of Intersection

The intersection of sets is important in searching for similarities between two or more sets. For example, consider two groups of friends: One of them enjoys reading books while the other gets some joy out of watching movies. The overlap of the two sets would be friends who love both books and movies belong to. They highlighted an intersection that also aims at a revelation of the commonalities of the groups. The parameters of intersection make the operation commutative: Commutative: $A \cap B=B \cap A$, associative: $(A \cap B) \cap C=A \cap(B \cap C)$, and the operation complies with the identity property with the universal set: $A \cap U=A$; which can ensure the consistency of the operation.

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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.

Set

Sets are a foundational concept in mathematics, central to various fields such as statistics, geometry, and algebra. 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…...

Intersection of sets

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.

Symbolically, we write $A \cap B=\{x: x \in A$ and $x \in B\}$
For example, let $A=\{2,4,6,8\}$ and $B=\{2,3,5,8\}$, then $A \cap B=\{2,8\}$

If $A$ and $B$ are two sets such that $A \cap B=\varphi$, then $A$ and $B$ are called disjoint sets.
For example, let $A=\{2,4,6,8\}$ and $B=\{1,3,5,7\}$. Then $A$ and $B$ are disjoint sets because there are no elements which are common to A and B .

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).

|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).

A intersection B formula
The $A$ intersection $B$ formula signifies the cardinality of a set. 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 $B . A=\{2,4,5$, $9,10,11,18,21\}, B=\{1,2,3,5,7,8,11,12,13\}$ 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 $A$ intersection $B$ which states: $n(A \cap B)=n(A)+$ $n(B)-n(A \cup B)$. We will 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) \\
& 3=8+9-14 \\
& 3=3
\end{aligned}
$

Hence, $\mathrm{n}(\mathrm{A}$ intersection B$)$ formula is verified.

Intersection of two sets
The intersection of two sets signifies that the common element is present in both sets.
For example, let $A=\{1,2,3,4\}$ and $B=\{3,4,5,6\}$, then $A \cap B=\{3,4\}$

Intersection of three sets

The intersection of three sets signifies that the common element is present in all three sets.
For example, let $A=\{2,4,6,8\}, B=\{2,3,8\}$, and $C=\{2,3,5,8\}$, then $A \cap B \cap C=\{2,8\}$
$A \cap B \cap C-\square$

Properties of intersection
- 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 laws
1. $A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$ i. e., $\cap$ distributes over $\cup$

This can be seen easily from the following Venn diagrams
LHS:
$A \cap(B \cup C)$

RHS :
$\left(A \cap_B\right)$
$(A \cap B) U(A \cap C)$

2. $A \cup(B \cap C)=(A \cup B) \cap(A \cup C) \mid$

This can be seen easily from the following Venn diagrams LHS:
$(B \cap C)$ $A \cup(B \cap C)$

RHS:
(AUB)
(AUC)
$(A \cup B) \cap(A \cup C)$

Summary

Equality of sets is one of the main topics in mathematics which has applications in real life in the sense of dividing the sets into the parity as discussed above. By knowing how the intersection is mathematically commutative, associative, and has an identity that is the Universal set, one can therefore relate one set of people to another. Whether it is learning what two friends have in common, deciding which records to show and which criteria out of a set of criteria.

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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.

.Frequently Asked Questions(FAQ)-

1. The comprehension of the cross-section set has been defined as the maximum over I of the sum of the coverage achieved by each subpopulation group.

Ans: The operation defined between two sets produces the set that embodies elements in common to the sets in question.

2. What is $\cap$ called?

Ans: The symbol $\cap$ represents the combined sets, in other words, the intersecting elements of the mentioned sets. For Example; in case where two sets $X$ and $Y$ are involved then; union of the set $=X \cup Y$ while intersection of the set $=X \cap Y$
3. In mathematics, set union and set intersection have different meanings and concepts; set union combines two sets while set intersection creates sets that have elements in common.

Ans: The union of sets $A$ and $B$ means that a set which contains all elements for which $t \in A$ or $t \in B$ is denoted as $A \cup B$ The intersection of two sets $A$ and $B$ means the set which contains in itself those elements which are present in both set A and set B . Writer also explains that if two sets are there, their overlapping portion is represented by $A \cap B$.
4. Another word for intersection:

Ans: The other symbol for intersection is 'AND,' so the symbol Venn diagram for the intersection of $A A B$ can also is written in form of $A A N D B$ which is equivalent to $A \cap B$.
5. Are $A \cap B$ and $B \cap A$ similar?

Ans: However, it goes without saying that $A \cap B=B \cap A$ which is a general property of an intersection operation - that is a commutative prodigy.