The intersection is an important topic in set theory, written with the symbol ∩.
It means taking only the elements that are common in two or more sets. This makes it very useful in mathematics, logic, and scientific writing.
In LaTeX, you can write this symbol in different ways, from a small intersection to big indexed intersections with limits.
Using the cap Command
The most direct way to write an intersection in LaTeX is with the \cap command. It represents the basic ∩ symbol used between two sets.
\documentclass{article}
\begin{document}
\[ s_{1}\cap s_{2} \]
\end{document}Using the bigcap Command
When the intersection involves several sets, a larger symbol is often needed. This is done using the \bigcap command.
\[ s_{1} \bigcap s_{2} \]Big Intersection with Limits
For multiple sets indexed with lower and upper limits, LaTeX provides \bigcap\limits and \bigcap with subscripts and superscripts.
These are useful for both finite and infinite collections of sets.
\bigcap\limits_{i=1}^{n}F_{i}
\bigcap_{i=1}^{n}A_{i}
\bigcap- This creates a large intersection symbol that works well for multiple sets in display math.
\limits- Forces the limits to appear directly above and below the symbol, which is useful in display style equations.
_{i=1}^{n}- This defines the lower limit (i=1) and the upper limit (n), showing that the intersection is taken across a sequence of sets.
\[ \bigcap\limits_{i=1}^{n}F_{i} \]
\[ \bigcap_{i=1}^{n}A_{i} \]Inline Big Intersection
When using the big ∩ symbol inside inline text, its size automatically adjusts. This keeps the equation aligned properly within a paragraph.
\( \bigcap_{i=1}^{n}x_{i} \)If you want the limits to appear below and above the symbol even in inline text, you need to use the \limits command. This forces LaTeX to place them exactly as in display mode.
\( \bigcap\limits_{i=1}^{n}x_{i} \)Custom Intersection with newcommand
Sometimes you may want a reusable intersection operator that always takes a lower and an upper limit.
This can be done with \newcommand. In this way you don’t have to write subscript and superscript again and again.
You can also keep \limits as an optional argument, so that in inline text you decide whether you want the limits above and below or not.
\newcommand{\Intersection}[3][ ]{\bigcap#1_{#2}^{#3}}
[]- The first argument is optional. If nothing is provided, the symbol behaves normally. If you pass
\limits, it will force the limits to appear above and below. {#2}- The second argument is required and is used as the lower limit.
{#3}- The third argument is required and is used as the upper limit.
\documentclass{article}
\usepackage{amsmath}
% Custom command with optional limits
\newcommand{\Intersection}[3][]{\bigcap#1\limits_{#2}^{#3}}
\begin{document}
% Normal display mode
\[
\Intersection{i=1}{n} A_i
\]
% Inline with side limits
Inline example: \( \Intersection{i=1}{n} A_i \)
% Inline with forced above-below limits
Inline with limits: \( \Intersection[\limits]{i=1}{n} A_i \)
\end{document}
Intersection Resulting in Empty Set
In many cases, intersections are performed on disjoint sets, which leads to an empty set. LaTeX uses the \emptyset command for this symbol.
\documentclass{article}
\begin{document}
\[ s_{1}\cap s_{2}=\emptyset \]
\end{document}For Infinite Sets
In advanced set theory, intersections are sometimes taken over infinitely many sets. In such cases, the upper limit is infinity, written as \infty.
\[ \bigcap_{i=1}^{\infty}A_{i} \]Best Practice
For standard use, \cap is enough, but when working with sequences of sets, \bigcap with lower and upper limits should be preferred.
Using \limits ensures clear alignment in display equations.
For infinite collections, the notation with \infty gives a precise mathematical meaning.
