--- aliases: [intersection, disjoint] --- #sets-functions ## Definition > [!tldr] Definition > The **intersection** of two [[Set|sets]] $A$ and $B$ is the set consisting of [[Set|elements]] that are in both $A$ and $B$. The notation for the intersection of $A$ and $B$ is $A \cap B$. > > In [[Set-builder notation|set-builder notation]], $A \cap B = \{x \, : \, x \in A \ \text{and} \ x \in B\}$. A visual representation of $A \cap B$ is this Venn diagram: ![[intersection.png]] **Notes:** - If $A$ and $B$ have no elements in common, then $A \cap B = \emptyset$ (the [[Empty set|empty set]]) and we say $A$ and $B$ are **disjoint**. ## Examples - If $A = \{1,2,3,4\}$ and $B = \{1,3,5,7,9\}$ then $A \cap B = \{3\}$. - If $A = \{n \in \mathbb{N} \, : \, n \ \text{is a multiple of 3}\}$ and $B = \{n \in \mathbb{N} \, : \, n \ \text{is a multiple of 2}\}$ then $A \cup B = \{0, 6, 12, 18, 24, \dots\}$ -- in other words all the natural numbers that are multiples of both $3$ and $2$ (which makes them multiples of $6$). ## Resources <iframe src="https://player.vimeo.com/video/606600971?h=e256db4c8a" width="640" height="360" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" allowfullscreen></iframe> <p><a href="https://vimeo.com/606600971">Screencast 3.6: Set operations</a> from <a href="https://vimeo.com/user132700952">Robert Talbert</a> on <a href="https://vimeo.com">Vimeo</a>.</p> Other resources: - Video: [Sets and set operations](https://www.youtube.com/watch?v=QiOfsWm3peE&list=PL2419488168AE7001&index=64&pp=iAQB) - Tutorial: [Intersection of sets](https://www.cuemath.com/algebra/intersection-of-sets/)