---
tag: 0HBRW4E6G1
kind: definition
name: Intersection
msc: ["03E20"]
defines: 0ZNMMA9TF5
uses: [0SJ3RX9TXA, 0EZ758XE73, 0M28ASWG8T]
sources:
  - url: https://en.wikipedia.org/wiki/Intersection_(set_theory)
    locator: lead section
---

## Definition

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
The intersection of $A$ and $B$ is the set
$$A \cap B = \{x \in A \mid x \in B\}.$$
So an object $x$ is an [[element|0EZ758XE73]] of $A \cap B$ when $x \in A$ and $x \in B$.

## Remarks

Example: $\{1, 2\} \cap \{2, 3\} = \{2\}$.
Intersection is commutative and associative, and $A \cap B \subseteq A$.
When $A \cap B = \varnothing$, the sets are [[disjoint|0W4RPA8QH3]].
The intersection of a family of sets is in [[intersection of a family of sets|0839XNQXDK]].

## Explanations

### L1 human

The intersection keeps only what the two sets have in common.
The intersection of the even numbers and the multiples of $3$ is the set of multiples of $6$.