Intersection of Sets — Definition, Formula & Examples

The intersection of sets is the set of all elements that belong to every one of the sets being compared. If an element appears in both Set A and Set B, it is in their intersection.

Given sets $A$ and $B$ , the intersection $A \cap B$ is defined as $\{x \mid x \in A \text{ and } x \in B\}$ . More generally, the intersection of a collection of sets contains exactly those elements that are members of every set in the collection.

Key Formula

A \cap B = \{x \mid x \in A \text{ and } x \in B\}

Where:

$A, B$ = The sets being intersected
$x$ = An element that belongs to both A and B
$\cap$ = The intersection symbol, read as 'intersect' or 'and'

How It Works

To find the intersection, list the elements of each set and identify which ones appear in all of them. Only elements shared by every set make it into the result. If no elements are shared, the intersection is the empty set

\varnothing

Worked Example

Problem: Let A = {2, 4, 6, 8, 10} and B = {3, 6, 9, 12, 8}. Find A ∩ B.

List elements of A: A contains 2, 4, 6, 8, and 10.

A = \{2, 4, 6, 8, 10\}

Check each against B: Go through each element of A and see if it also appears in B. The number 6 is in B, and 8 is in B. The others (2, 4, 10) are not.

6 \in B, \quad 8 \in B

Write the intersection: Collect the shared elements into a new set.

A \cap B = \{6, 8\}

Answer:

A \cap B = \{6, 8\}

Why It Matters

Intersection shows up whenever you need to find what two groups have in common — students who play both soccer and basketball, numbers that are both even and prime, or survey respondents who chose multiple options. It is a core operation in Venn diagram problems and a building block for probability, where

P(A \cap B)

represents the chance that two events both occur.

Common Mistakes

Mistake: Confusing intersection (∩) with union (∪) and listing all elements from both sets instead of only the shared ones.

Correction: Intersection means 'and' — include only elements found in every set. Union means 'or' — that is when you combine all elements.