Set Symbols Explained: Union ∪, Intersection ∩, Subset ⊆, Element ∈, and More

Set notation uses a small family of symbols to describe collections of objects: ∪ (union) joins two sets, ∩ (intersection) keeps only what they share, ⊂ and ⊆ say one set sits inside another, ∈ means "is an element of," ∉ means "is not," and ∅ is the empty set. The number sets have their own double-struck letters: ℕ for the natural numbers, ℤ for the integers, ℚ for the rationals, ℝ for the reals, and ℂ for the complex numbers. In LaTeX the commands are \cup, \cap, \subset, \subseteq, \in, \notin, \emptyset, and \mathbb{R}.

This guide lists every common set symbol with its Unicode code point, explains what each one means with a Venn-diagram picture in mind, and shows how to type them on Windows, Mac, phones, Google Docs, Word, HTML, and LaTeX.

Copy and Paste the Set Notation Symbols

What the Set Symbols Mean

A set is a collection of distinct objects, written with curly braces: A = {1, 2, 3}. The symbols above describe how sets relate to each other and to their elements. Picture two overlapping circles, A on the left and B on the right, inside a rectangle U that holds everything under discussion; that is the Venn diagram every definition below refers to.

Membership: ∈ and ∉

x ∈ A says x is one of the objects in A, and x ∉ A says it is not. With A = {1, 2, 3}: 2 ∈ A, but 5 ∉ A. Membership is about elements, not sets, so {1} ∈ {1, 2} is false even though 1 ∈ {1, 2} is true.

Union and Intersection: ∪ and ∩

A ∪ B is both circles shaded together: everything in A, everything in B, with shared elements listed once. A ∩ B is the lens where the circles overlap: only the elements in both. If A = {1, 2, 3} and B = {3, 4}, then A ∪ B = {1, 2, 3, 4} and A ∩ B = {3}. When the overlap is empty, A ∩ B = ∅ and the sets are called disjoint. The symbols are deliberately shaped like the logical ∨ (or) and ∧ (and), because x ∈ A ∪ B means x ∈ A or x ∈ B, and x ∈ A ∩ B means x ∈ A and x ∈ B.

Subsets: ⊆, ⊂, ⊊, and ⊃

A ⊆ B says every element of A is also in B: the A circle lies entirely inside the B circle. The line under the symbol permits A = B, in the same way ≤ permits equality. A ⊊ B adds that A is strictly smaller. The bare ⊂ is ambiguous: many textbooks use it for "proper subset," while many research papers use it to mean ⊆. If the distinction matters, write ⊆ or ⊊ and avoid ⊂ altogether. ⊇ and ⊃ are the same relations read from the larger set. Every set is a subset of itself, and ∅ is a subset of every set.

Empty Set: ∅

∅ is the set with nothing in it, {}. It is not the number zero and not the digit Ø, even though Windows Alt + 0216 produces the Scandinavian letter Ø that people often paste in its place. Unicode's empty set is U+2205. Two forms appear in LaTeX: \emptyset gives a narrow, slightly squashed glyph, and \varnothing from amssymb gives a round circle with a slash that most authors find nicer.

Set Difference, Complement, and Product: ∖, ∁, ×

A ∖ B (also written A − B) is the part of the A circle outside the B circle. The complement Aᶜ, A′, or ∁A is everything in the rectangle U that is not in A, which is U ∖ A. The symmetric difference A △ B is both circles minus the lens. The Cartesian product A × B is a different kind of object: the set of all ordered pairs (a, b) with a ∈ A and b ∈ B, so {1, 2} × {x, y} = {(1, x), (1, y), (2, x), (2, y)}, and ℝ × ℝ = ℝ² is the coordinate plane.

Set-Builder Notation

{x ∈ ℝ | x > 0} is read "the set of all x in the reals such that x is greater than zero." The vertical bar (or a colon) separates the variable from its condition. The number sets are explained in more depth at https://www.useoctree.com/blog/real-numbers-symbol-blackboard-bold, and the "such that" bar is best typed as \mid in LaTeX so that it gets relation spacing.

De Morgan's Laws

Two identities tie the symbols together and are worth memorising because they are the set version of negating "and" and "or":

• (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ — not in either means in neither
• (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ — not in both means missing from at least one

Shade the Venn diagram for each side and the pictures match.

How to Type Set Symbols

Windows: Only ∩ has a classic Alt code (Alt + 239, from code page 437). For the rest, type the hex value in Word and press Alt + X: 222A becomes ∪, 2208 becomes ∈, 2205 becomes ∅, 211D becomes ℝ. Holding Alt and typing the decimal Unicode value (8746 for ∪, 8712 for ∈) on the numeric keypad works in Word and WordPad but not in most other programs. Insert → Symbol with the subset set to Mathematical Operators shows the whole family together, and the double-struck letters are under Letterlike Symbols.

Mac: There are no Option-key shortcuts for set symbols. Open the Character Viewer with Control + Command + Space, search "union", "intersection", "element", "subset", "empty set", or "double-struck", and double-click. For frequent use, add text replacements in System Settings → Keyboard, mapping for example ;in to ∈ and ;cup to ∪.

iPhone and Android: Stock keyboards offer none of these. Copy them from the table above, or create text replacements (iOS: Settings → General → Keyboard → Text Replacement; Gboard: personal dictionary). A dedicated math keyboard app is another option.

Google Docs and Word: In Google Docs, Insert → Special characters lets you search by name or draw the symbol. In Word, use Insert → Symbol or the Alt + X trick. Both equation editors accept LaTeX-style names: typing \cup, \in, \subseteq, or \emptyset followed by a space converts to the symbol, and \doubleR in Word gives ℝ.

Set Notation in LaTeX

\cup, \cap, \subset, \subseteq, \supset, \supseteq, \in, \notin, \ni, \emptyset, \setminus, and \times are all in the LaTeX kernel. The big operators \bigcup and \bigcap take limits for unions and intersections over an index. \mathbb{R} needs amssymb (or the lighter amsfonts), which also provides \varnothing, \subsetneq, \nsubseteq, \complement, and \smallsetminus. For set-builder notation, \{ x \in \mathbb{R} \mid x > 0 \} gives the braces and the properly spaced bar, and \left\{ ... \right\} scales them to a tall condition.

1\documentclass{article}
2\usepackage{amsmath}
3\usepackage{amssymb}
4\begin{document}
5Membership and subsets: $2 \in A$, $5 \notin A$, $A \subseteq B$,
6$A \subsetneq B$, $C \not\subset A$, and $\emptyset \subseteq A$.
7
8Operations: $A \cup B$, $A \cap B = \varnothing$, $A \setminus B$,
9$A \times B$, $\complement A$, and $A \triangle B$.
10
11Number sets and set-builder notation:
12\[ \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C},
13   \qquad \{\, x \in \mathbb{R} \mid x > 0 \,\} \]
14
15Big operators and De Morgan:
16\[ \bigcup_{i=1}^{n} A_i, \qquad \bigcap_{i=1}^{n} A_i, \qquad
17   (A \cup B)^{c} = A^{c} \cap B^{c} \]
18\end{document}
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All of these commands are math-mode only, so wrap them in $...$ in a sentence. The thin spaces \, inside the braces are optional but make set-builder notation easier to read. For the complete set-theory table with code points, see https://tools.useoctree.com/symbols/set-theory, and for math mode basics see https://www.useoctree.com/learn/mathematical-expressions.

FAQ

What is the difference between ⊂ and ⊆?

⊆ means "subset of or equal to": A ⊆ B allows A and B to be the same set. ⊂ is used by some authors for "proper subset" (A is inside B but not equal to it) and by others as a synonym for ⊆. To be unambiguous, write ⊆ for the inclusive relation and ⊊ for the strict one.

What does ∈ mean?

∈ means "is an element of" or "belongs to": 3 ∈ {1, 2, 3}. The crossed-out form ∉ means "is not an element of." The symbol is a stylised Greek epsilon, chosen by Peano in 1889 from the Italian word è, "is."

What is the symbol for the empty set?

∅ (U+2205), read "empty set" or "null set," and written {} in braces notation. It is not the same as the letter Ø or the number 0. In LaTeX use \emptyset or, with amssymb, the rounder \varnothing.

How do I type the union and intersection symbols?

In LaTeX write \cup for ∪ and \cap for ∩ (think "cup" and "cap" for their shapes). In Word type 222A or 2229 and press Alt + X. In HTML use ∪ and ∩. On a Mac, search the Character Viewer; on phones, copy and paste.

What do ℕ, ℤ, ℚ, ℝ, and ℂ stand for?

They are the number sets in blackboard bold: ℕ natural numbers, ℤ integers (from the German Zahlen), ℚ rationals (quotients), ℝ reals, and ℂ complex numbers. Each one contains the previous one, ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ.

Writing Set Notation Cleanly in Your Documents

Use \subseteq when equality is allowed and \subsetneq when it is not, \mid for the "such that" bar, and \varnothing for a readable empty set. Octree, the AI LaTeX editor at https://www.useoctree.com, autocompletes every set symbol and previews the compiled result instantly, so your definitions typeset exactly the way you mean them.