Editing Colon (punctuation) (section)

==Mathematics and logic<!--'Such that' redirects here-->==
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The colon is used in [[mathematics]], [[cartography]], [[scale model|model building]], and other fields, in this context it denotes a [[ratio]] or a [[scale (ratio)|scale]], as in 3:1 (pronounced "three to one").<ref name="PunctuationGuide"/>

When a ratio is [[reduction (mathematics)|reduced]] to a simpler form, such as 10:15 to 2:3, this may be expressed with a [[double colon (disambiguation)|double colon]] as 10:15::2:3; this would be read "10 is to 15 as 2 is to 3". This form is also used in tests of logic where the question of "Dog is to Puppy as Cat is to _____?" can be expressed as "Dog:Puppy::Cat:_____". For these uses, there is a dedicated [[Unicode]] symbol ({{unichar|2236}}) that is preferred in some contexts. Compare {{not a typo|2:3}} (ratio colon) with 2:3 (U+003A ASCII colon).

In some languages (e.g. German, Russian, and French), the colon is the commonly used sign for division (instead of ÷).
{{Crossreference|(See also [[Division sign]] and {{section link|Division (mathematics)|Notation}}.)}}

The notation |{{mvar|G}} : {{mvar|H}}| may also denote the [[index of a subgroup]].

The notation {{math|ƒ : {{mvar|X}} → {{mvar|Y}}}} indicates that {{mvar|f}} is a [[function (mathematics)|function]] with domain {{mvar|X}} and codomain {{mvar|Y}}.

The combination with an equal sign ({{math|≔}}) is used for [[definition]]s.

In [[mathematical logic]], when using [[set-builder notation]] for describing the characterizing property of a [[set (mathematics)|set]], it is used as an alternative to a [[vertical bar]] (which is the [[ISO 31-11]] standard), to mean "'''such that'''". Example:
:<math>S = \{x \in \mathbb{R} : 1 < x < 3 \}</math> (''S'' is the set of all {{mvar|x}} in <math>\mathbb{R}</math> (the [[real number]]s) such that {{mvar|x}} is strictly greater than 1 and strictly smaller than 3)

In older literature on mathematical logic, it is used to indicate how expressions should be bracketed (see [[Glossary of Principia Mathematica|Glossary of ''Principia Mathematica'']]).

In [[type theory]] and [[programming language theory]], the colon sign after a term is used to indicate its type, sometimes as a replacement to the "∈" symbol. Example:
:<math>\lambda x . x \mathrel{:} A \to A </math>.

A colon is also sometimes used to indicate a [[tensor contraction]] involving two indices, and a double colon (::) for a contraction over four indices.

A colon is also used to denote a [[parallel sum]] operation involving two operands (many authors, however, instead use a [[∥]] sign and a few even a [[∗]] for this purpose).