Set & logic symbols

The symbols of set theory and logic — quantifiers like ∀ and ∃, the connectives ∧, ∨, ¬, set symbols such as ∈, ⊂, ∪, and the proof marks ⊢, ⊨, ∴, ∵ — are all entered inside math mode as commands such as \forall, \in, and \cup. Three things are worth fixing in your mind: most are available in standard LaTeX; \land/\lor are aliases of \wedge/\vee; and a handful, including \nexists, \complement, and \therefore, need the amssymb package. This page sorts out how to enter them, gathers quantifiers & logic, sets, proof (turnstiles), and reasoning marks into lookup tables, and clears up confusable choices such as \varnothing versus \emptyset.

Entering the symbols, and packages

These symbols are used inside math mode. Writing \forall straight into the body (text mode) is an error, so you enter math mode first, as in $\forall x$. The command names mirror their English meaning — \forall, \exists, \in (element of), \subset (subset), \cup (union), and so on.

\[
  \forall \varepsilon > 0 \;\exists \delta > 0 \;
  \bigl( |x - a| < \delta \implies |f(x) - f(a)| < \varepsilon \bigr)
\]
\[
  A \cup B = \{\, x : x \in A \lor x \in B \,\}, \qquad
  A \subseteq B \iff (\forall x)\,(x \in A \Rightarrow x \in B)
\]

Here \forall, \exists, \in, \cup, \subseteq, and \Rightarrow are standard LaTeX and need no extra package. By contrast \implies and \iff come from amsmath, while \nexists, \varnothing, \complement, \therefore, and \because (all below) require amssymb. In practice, loading both amsmath and amssymb in the preamble makes every symbol on this page available.

\usepackage{amsmath}  % \implies, \iff（間隔つきの長い矢印）
\usepackage{amssymb}  % \nexists, \varnothing, \complement, \therefore, \because

In the tables below, anything needing amssymb is marked "(ams)"; treat unmarked commands as standard LaTeX. Symbols also carry a class: \in and \subset are relations (a touch more space on each side), \cup, \cap, \land, \lor are binary operators (symmetric spacing), and \forall, \neg, \top set as ordinary symbols. This class is what governs the spacing LaTeX inserts automatically.

Quantifiers and logical connectives

First, the backbone of predicate logic. The universal ∀ (\forall) and existential ∃ (\exists), with the negated “does not exist” ∄ (\nexists, ams). The connectives come down to five: negation ¬, conjunction ∧, disjunction ∨, implication ⇒, and equivalence ⇔. Implication and equivalence have two forms: \Rightarrow/\Leftrightarrow are the short double-line arrows, while \implies/\iff (amsmath) are longer arrows with wider space on each side, easier to read when you write a chain of reasoning that reads almost like prose.

A few cautions. **\neg and \lnot are exactly the same**, and \land/\lor are merely aliases of \wedge/\vee (no \amssymb needed). The \land/\lor names read clearly as logic and suit propositional formulas; \wedge/\vee are preferred where the meaning is not “logic” — wedge products, lattice meet/join, and the like. The output is identical, so choose by taste and context. \implies/\iff are really \Longrightarrow/\Longleftrightarrow with a thick space (a \;) on each side; \iff is a LaTeX-kernel command that amsmath redefines to improve that spacing. Keep your use of the short \Rightarrow versus the long \implies consistent within a document.

Set symbols

Now the symbols around sets. The empty set comes in two shapes: the standard \emptyset (∅) and amssymb’s \varnothing (∅). The latter is a clean circle crossed by a diagonal slash, and **many people prefer \varnothing** over the taller-looking \emptyset, so we list it as the recommended form. Membership is ∈ (\in) with its negation ∉ (\notin), and the mirrored ∋ (\ni, “owns / has as element”). Containment runs from proper subset ⊂ and subset-or-equal ⊆ to their reverses ⊃ and ⊇. The set operations are union ∪, intersection ∩, difference ∖ (\setminus), and complement ∁ (\complement, ams).

Some guidance. For the empty set, **make \varnothing (amssymb) your default and be consistent** throughout a document. Whether \subset means “proper subset” or “subset or equal” varies by field, so to avoid ambiguity use ⊆ (\subseteq), or the dedicated proper-subset ⊊ (\subsetneq, amssymb). Set difference is \setminus (∖), not the division slash /. The complement is often written with \complement as a superscript, A^\complement, though \overline{A} (a bar) or A^c are also common.

Proof and turnstile symbols

These mark “derivable” and “satisfies” in logic and proof theory. ⊢ (\vdash) is the turnstile for syntactic provability (“Γ ⊢ φ: φ is provable from Γ”); ⊨ (\models) is the double turnstile for semantic entailment (“Γ ⊨ φ: Γ entails / satisfies φ”). ⊣ (\dashv) is ⊢ reflected left-to-right, also used in notation for adjoint functors. ⊤ (\top) is verum / the greatest element and ⊥ (\bot) is falsum / the least element (top and bottom of a lattice or order); \bot doubles as the symbol for “perpendicular / orthogonal.”

These are all standard LaTeX — no extra package needed. \vdash and \models are relations, so they get appropriate space on each side and set naturally in two-sided forms like Γ \vdash φ. Note that ⊤ and ⊥ are ordinary symbols, so using them as if they were binary relations can look a little tight in spacing.

Reasoning marks (∴, ∵)

Finally, two marks that signal the flow of an argument. ∴ (\therefore, “therefore”) introduces a conclusion, and ∵ (\because, “because”) introduces a reason. **Both require amssymb** — without \usepackage{amssymb} in the preamble you get an “undefined control sequence” error.

% プリアンブル: \usepackage{amssymb}
\[
  x^2 = 4 \quad \therefore\ x = \pm 2,
  \qquad x = \pm 2 \quad \because\ x^2 = 4
\]

Because these set as ordinary symbols, they do not get much space around them automatically. Adding explicit spacing (\ or \quad), as in \therefore\ above, makes them read better. They are common on the blackboard, but polished mathematical prose often prefers words like “therefore” and “hence,” so ∴ and ∵ are symbols to use judiciously.