For All (∀) and There Exists (∃) Symbols: Meaning, Uses and How to Type Them

The for all symbol ∀ is an upside-down A, and the there exists symbol ∃ is a backwards E. Together they are the quantifiers of logic: ∀x P(x) says that P holds for every x, and ∃x P(x) says that P holds for at least one x. They let a definition such as "every real number has a square that is not negative" fit on one line: ∀x ∈ ℝ, x² ≥ 0.

Below: the characters to copy, the codes for every platform, how to read and negate quantified statements, and how to write them in LaTeX.

For All and There Exists Symbols: Copy and Paste

Copy the symbols from the box below, or take the code you need from the table:

1∀ ∃ ∄
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For All and There Exists Definition

∀ is the universal quantifier: ∀x P(x) is true when P(x) is true for every x in the domain. ∃ is the existential quantifier: ∃x P(x) is true when at least one x makes P(x) true. A quantifier usually names its domain, as in ∀x ∈ ℝ or ∃n ∈ ℕ, and the statement after it is often separated by a comma, a colon or a dot.

Order matters. ∀x ∃y (y > x), "for every number there is a bigger one," is true for the integers. ∃y ∀x (y > x), "there is a number bigger than every number," is false.

The two are linked by negation, one of the most useful rules in proofs:

• ¬∀x P(x) ≡ ∃x ¬P(x): "not everything is P" means "something is not P."
• ¬∃x P(x) ≡ ∀x ¬P(x): "nothing is P" means "everything is not P," also written ∄x P(x).

Giuseppe Peano used a reversed E for existence in 1897, and Gerhard Gentzen introduced the turned A for "for all" in 1935, by analogy with ∃.

Uses of the Quantifier Symbols

• Definitions in analysis: the limit definition reads ∀ε > 0 ∃δ > 0 such that 0 < |x − a| < δ ⇒ |f(x) − L| < ε. The order of ∀ε and ∃δ is exactly what the definition says.
• Set theory: A ⊆ B means ∀x (x ∈ A ⇒ x ∈ B), and a set-builder expression like {x ∈ ℤ : ∃k ∈ ℤ, x = 2k} is the even integers. See set symbols.
• Uniqueness: ∃! x P(x) means exactly one x satisfies P, as in "∃! x ∈ ℝ such that x + 3 = 5."
• Computer science: formal specifications and type theory use ∀ for polymorphism (∀a. a → a is the type of the identity function), and SQL's EXISTS keyword is the database version of ∃.

How to Type the For All and There Exists Symbols

• Windows: no Alt codes exist for ∀ or ∃. In Word, type 2200 or 2203 and press Alt + X, or insert an equation (Alt + =) and type \forall or \exists and a space.
• Mac: open the Character Viewer (Control + Command + Space) and search "for all" or "exists," or use Unicode Hex Input and hold Option while typing 2200.
• Google Docs: Insert → Special characters and search "for all," or type \forall in an equation.
• HTML: ∀, ∃ and ∄.
• Phones: no default keyboard has them; copy from this page or set a text replacement.

For All and There Exists in LaTeX

\forall and \exists are built into LaTeX. \nexists needs amssymb. Spacing after a quantifier is not automatic, so add a thin space or a comma.

1\documentclass{article}
2\usepackage{amsmath, amssymb}
3\begin{document}
4\[ \forall x \in \mathbb{R},\ x^2 \geq 0 \]
5
6\[ \forall \varepsilon > 0 \ \exists \delta > 0 :\
7   0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon \]
8
9Negation: $\neg \forall x\, P(x) \iff \exists x\, \neg P(x)$
10
11Uniqueness: $\exists!\, x \in \mathbb{R} : x + 3 = 5$
12
13None: $\nexists\, x \in \mathbb{Q} : x^2 = 2$
14\end{document}
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• \, (thin space) after the bound variable and \ or \quad between quantifiers keep long formulas readable.
• \mathbb{R} needs amssymb (or amsfonts); \neg gives ¬, \implies and \iff come from amsmath.
• If \nexists is unavailable, \not\exists gives a similar slashed symbol.

All logic and quantifier symbols with their commands are on the quantifier symbol list, and the math symbols lesson lists the rest of the logic notation. To mark the conclusion of an argument, see the therefore symbol.

Related symbols and guides

• For All symbol page — copy the symbol and its LaTeX command
• Exists symbol page — copy the symbol and its LaTeX command
• Quantifier symbols reference — every symbol in the category with its LaTeX command
• Mathematical expressions in LaTeX — how to typeset these symbols in a document
• Therefore symbol (∴)

For All and There Exists Symbols FAQ

What does the upside-down A mean in math?

The upside-down A, ∀, means "for all" or "for every." ∀x P(x) says the statement P(x) is true for every x in the domain.

What does the backwards E mean in math?

The backwards E, ∃, means "there exists" or "for some." ∃x P(x) says at least one x makes P(x) true. ∃! means "there exists exactly one."

How do you negate ∀ and ∃?

Swap the quantifier and negate what follows: ¬∀x P(x) is equivalent to ∃x ¬P(x), and ¬∃x P(x) is equivalent to ∀x ¬P(x).

How do I type ∀ and ∃ in LaTeX?

Use \forall and \exists in math mode, e.g. $\forall x \in A$. For ∄ use \nexists with amssymb.

How do I type the for all symbol in Word?

Type 2200 and press Alt + X (2203 for ∃), or insert an equation with Alt + = and type \forall followed by a space.