Relation symbols

A relation symbol sits between two expressions and asserts a relationship — “equal,” “less than or equal,” “is a member of,” and so on. =, \leq, and \in are all relations. Inside TeX, relations form a single symbol class that automatically gets a wide space on each side — wider than the space around binary operators. This page first fixes the idea of a relation *as a class*, then gathers the symbols into lookup tables grouped by use — order/comparison, similarity, order-theoretic, set relations, geometry and others. It marks which commands are plain LaTeX, which need amssymb, and how to negate a relation with \not.

Relations as a symbol class

TeX sorts every symbol in a formula into a handful of classes and chooses the surrounding space from the class. The binary-operator class (+, -, \times) gets a medium space, while the relation class (=, <, \leq) gets a wider one. That is why a=b has more room around its sign than a+b does around the plus. You do not insert this space yourself with \, and friends — it follows automatically from the symbol’s class.

A relation is also something you place *between* two expressions. Chain them as in a \leq b \leq c and each relation gets equal space on both sides. Conversely, to typeset a symbol of your own — or one built with \mathbf — *as* a relation, wrap it in \mathrel{...} so it joins the relation class and picks up the right spacing (\mathbin{...} does the same for a binary operator). The full picture of symbol classes is covered under “Math mode basics.”

\[ a \leq b < c, \qquad x \equiv y \pmod{n} \]
% 自作・既存の記号を「関係子」として組む（左右に関係子の空きが付く）
\[ A \mathrel{R} B, \qquad x \mathrel{\sim_{\!\ast}} y \]

One caution: arrows (\to, \rightarrow, …) also belong to the relation class internally, yet by look and purpose they form a separate family. In particular \to is an arrow meaning “maps to,” not a comparison relation like \leq. Arrows are collected on the “Arrows” page. To compare the relation space against the narrower binary-operator space, see the “Binary operators” page as well.

Order and comparison

The most-used relations express size and (in)equality. < and > are characters you type directly in math mode, so they need no command, but “≤” and “≥” are \leq and \geq. These have short aliases \le and \ge that produce exactly the same symbol. “Not equal” is \neq (alias \ne). Everything in the table below is plain LaTeX (no package needed).

Typing < or > straight into the body (text mode) can turn into a different character, so use the inequality signs inside math mode. \equiv (≡) serves both “congruent” (integer congruence a \equiv b \pmod n) and “identically equal,” the sense fixed by context. \doteq (≐) is sometimes read “equal by definition,” but for a clean := reach for mathtools’ \coloneqq (below).

Similarity and approximation

This group expresses relations looser than equality: “approximately equal,” “similar to,” “proportional to.” The tilde-family symbols \sim (∼), \simeq (≃), \approx (≈), and \cong (≅) look alike, so rely on the command name to keep them apart. Proportionality is \propto (∝). These are plain LaTeX too.

As a rule of thumb: \approx (≈) for numeric approximation (\pi \approx 3.14); \sim (∼) for an equivalence relation or “same order”; \simeq (≃) for homeomorphism or asymptotic equivalence; and \cong (≅) for geometric congruence or algebraic isomorphism. Finer approximation symbols (\lesssim, \gtrsim, \approxeq, …) live in amssymb (next section).

Order-theoretic relations

Partial orders and precedence are written with the prec/succ family (precede/succeed). There are \prec (≺), \succ (≻), and the “or-equal” forms \preceq (⪯), \succeq (⪰). Use them for an order distinct from numeric size (\leq). These are plain LaTeX as well.

Their negations \nprec, \nsucc, \npreceq, \nsucceq, the tilde forms \precsim, \succsim, and the curly \preccurlyeq are provided by amssymb. To negate with plain LaTeX only, prefix \not (below) — though the resulting glyph is not ideal.

Set relations

This group expresses set inclusion and membership. \subset (⊂) and \supset (⊃) are (proper) subset and superset; the “or-equal” forms \subseteq (⊆) and \supseteq (⊇) mean “subset-or-equal.” Membership is \in (∈, “is an element of”) and its mirror \ni (∋, “owns / has as a member”; alias \owns). “Not a member” has a dedicated command \notin (∉). All are plain LaTeX.

Two cautions. First, whether \subset (⊂) means “proper subset” or just “subset” varies by field and author, so state your convention in the document if it matters. Second, the negation \nsubseteq (⊈), the explicit-proper \subsetneq, \supsetneq, and the like come from amssymb. Only \notin (∉) has a dedicated command in plain LaTeX — unlike \nsubseteq and the others.

Geometry and other relations

These relations show up in geometry and number theory: parallel and perpendicular, and the vertical bar for divisibility. \parallel (∥, parallel; alias \|), \perp (⊥, perpendicular), and \mid (∣, a single vertical bar) for “divides.” These are plain LaTeX. \smile (⌣) and \frown (⌢) are also standard, but note that in TeX’s class scheme they are ordinary symbols, not relations — so they do not get the wide relation space on each side.

\mid (∣) also serves as the “such that” bar in set-builder notation \{\, x \mid x > 0 \,\}, with better spacing than a bare |. The negation of parallel, \nparallel (∦), and the “does not divide” symbol \nmid (∤) are provided by amssymb.

amssymb and how to negate

Every symbol above works in plain LaTeX, but to add more relations the standard move is the **amssymb package** (\usepackage{amssymb} in the preamble). Two especially common additions are the slanted variants of the inequality signs, \leqslant (⩽) and \geqslant (⩾), and the negated forms of many relations — dedicated symbols with a slash already drawn through them.

To negate without adding a package, put **\not** before the relation: \not= (means ≠), \not\leq, \not\subset — you can slash through any relation. But \not’s slash has a fixed size, slope, and position, so it can sit awkwardly over some symbols. Hence the rule of thumb: when a dedicated negation exists, prefer amssymb’s \nleq, \nsubseteq, and so on for a cleaner glyph. (\neq (≠) and \notin (∉) already have dedicated commands in plain LaTeX, so they need no \not.)

\usepackage{amssymb}   % \leqslant, \nleq, \nsubseteq …
\usepackage{mathtools}  % \coloneqq (:=)
% ...
% 斜めの不等号と、専用の否定記号
\[ 0 \leqslant x \leqslant 1, \qquad a \nleq b, \qquad A \nsubseteq B \]
% \not による即席の否定（専用記号がないとき）
\[ x \not\equiv y \pmod{p} \]
% 「定義により等しい」は mathtools の \coloneqq が綺麗
\[ f(x) \coloneqq x^2 + 1 \]

One more common need is the definition sign :=. Typed directly, the colon sits low against the equals, so the mathtools package’s **\coloneqq** (:=) lines them up cleanly (its mirror \eqqcolon gives =:). Note that \coloneqq comes from mathtools, not amssymb. For the broader set of symbols amssymb unlocks, see the “amsmath / AMSFonts” page.