Intersection

For two sets, write $A\cap B=\bigcap\{A,B\}$. For an indexed family $(A_i)_{i\in I}$, the notation $\bigcap_{i\in I}A_i$ means the intersection of the sets in its range.

An empty intersection requires an ambient set $X$: for subsets $A_i\subseteq X$, set $\bigcap_{i\in\varnothing}A_i=X$. Without an ambient set, there is no set of all sets.

Intersection is dual to union and is closely related to containment: one has $A\subseteq B$ exactly when $A\cap B=A$.

• $\{1,2\}\cap\{2,3\}=\{2\}$.
• If $A=\{x\in\mathbb{R}: x<0\}$ and $B=\{x\in\mathbb{R}: x\ge 0\}$, then $A\cap B=\varnothing$ (see empty set).