Intersection of Sets

Intersection of Sets - Definition, Explanation, and Examples

The intersection of two sets is an important set operation that finds all elements common to both sets.

Definition of Intersection

The intersection of two sets $A$ and $B$ is the set containing all elements that are in both $A$ and $B$. It is denoted by:

$A\cap B$

Mathematically:

$A\cap B=\{x:x\in A\text{ and }x\in B\}$

Understanding Intersection

When we take the intersection of two sets, we look for elements that appear in both sets simultaneously.

Step-by-Step Explanation

1. Identify all elements of set $A$. 2. Identify all elements of set $B$. 3. Find the elements that are present in both $A$ and $B$. 4. Form a new set with these common elements.

Examples of Intersection of Sets

Example 1: Numbers

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

Step 1: Elements of $A$: 1, 2, 3, 4. Step 2: Elements of $B$: 3, 4, 5, 6. Step 3: Common elements: 3, 4. Step 4: Intersection: $A\cap B=\{3,4\}$

Example 2: Letters

Let $C=\{\text{a},\text{b},\text{c}\}$ $D=\{\text{b},\text{d},\text{e}\}$

Step 1: Elements of $C$: a, b, c. Step 2: Elements of $D$: b, d, e. Step 3: Common elements: b. Step 4: Intersection: $C\cap D=\{b\}$

Example 3: Students

Class 1 students: $E=\{\text{John},\text{Emma},\text{Liam}\}$ Class 2 students: $F=\{\text{Emma},\text{Olivia},\text{Noah}\}$

Step 1: Elements of $E$: John, Emma, Liam. Step 2: Elements of $F$: Emma, Olivia, Noah. Step 3: Common element: Emma. Step 4: Intersection: $E\cap F=\{\text{Emma}\}$

Summary

• The intersection of two sets contains only the elements common to both.
• If there are no common elements, the intersection is the empty set $\mathrm{\varnothing }$.
• The intersection helps in finding shared characteristics or common data points.