Intersection of Sets

What is the Intersection of Sets? 

The intersection of sets is a binary operation that finds the common elements between two or more sets. The resulting set contains only the elements that are present in all the original sets. Think of it as finding the common ground or the overlap between different groups.

1.0Intersection of Sets Definition

Formally, the intersection of two sets, say set A and set B, is a set that consists of all the elements that belong to set A and set B simultaneously.

• Intersection of sets symbol: The symbol used to denote the intersection of sets is '∩'. So, the intersection of sets A and B is written as A∩B.
• Set-Builder Notation: A∩B = {x∣x∈A and x∈B}

This notation reads: "A intersection B is the set of all elements x such that x is an element of A and x is an element of B."

2.0Intersection of Sets Symbol

The intersection of sets symbol is “($\cap$)”, which resembles an upside-down “U”. This is standard notation in mathematics.

Example: If ( A = {1, 2, 3} ) and ( B = {2, 3, 4} ), then ( $A\cap B$ ) denotes their intersection.

3.0Intersection of Sets Formula

The intersection of sets formula is vital for calculating the number of common elements, especially in problems involving the principle of inclusion-exclusion.

Formula for Two Sets

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

Where:

• ( $n(A\cap B)$ ) = Number of elements common to A and B
• ( n(A) ) = Number of elements in set A
• ( n(B) ) = Number of elements in set B
• ( $n(A\cup B)$ ) = Number of elements in the union of A and B

Formula for Three Sets

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

This is an extension of the inclusion-exclusion principle, commonly tested in JEE problems involving three or more sets.

4.0Intersection of Sets Example

Let’s understand with some practical examples:

Example 1: Let ( A = {2, 4, 6, 8} ), ( B = {4, 8, 12, 14} ).

• Step 1: List the elements of both sets.
• Step 2: Identify the common elements.

$A\cap B=\{4,8\}$

Example 2: If ( $X=\text{{}a,b,c\}$ ) and ( $Y=\text{{}b,c,d,e\}$ ):

$X\cap Y=\text{{}b,c\}$

5.0Intersection of Sets Venn Diagram

The intersection of two sets can be represented using the Venn diagram as shown below.

In the above diagram, the shaded portion represents the intersection of two sets A and B.

Similarly, we can draw a Venn diagram for the intersection of 3 sets as shown below.

In the above diagram, we can see that the centermost region denotes the intersection of three sets A, B and C.

6.0Intersection of Sets Diagram Representation

Sets can be represented in various forms—roster form, set-builder notation, or visually, such as with Venn diagrams.

Roster Form Example

If ( A = {1, 3, 5, 7} ), ( B = {5, 7, 9} ): $A\cap B=\{5,7\}$

Set-Builder Notation Example

$A=\{x:x\text{ is an odd integer less than 8}\}$

$B=\{x:x\text{ is an odd integer greater than 4 and less than 10}\}$

$A\cap B=\{x:x\text{ is an odd integer such that }4<x<8\}=\{5,7\}$

7.0Properties of Intersection of Sets

The properties of intersection of sets are important for solving problems and understanding set relationships.

1. Commutative Law: $A\cap B=B\cap A$

• Order does not affect the result.

2. Associative Law: $(A\cap B)\cap C=A\cap (B\cap C)$

• Grouping does not affect the outcome.

3. Identity Law: $A\cap U=A$, where ( U ) is the universal set.
4. Idempotent Law: $A\cap A=A$

• Intersection of a set with itself is the set itself.

5. Domination Law: $A\cap \varnothing =\varnothing$

• Intersection of any set with the empty set is the empty set.

6. Subset Law: If ( $A\subseteq B$ ), then ( $A\cap B=A$ ).
7. Distributive Law: $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$
8. De Morgan’s Law : $(A\cap B{)}^{\prime }=A^{\prime }\cup B^{\prime }$, where the prime denotes the complement.