Colon (punctuation) - Mathematics and Logic

Mathematics and Logic

The colon is used in mathematics, cartography, model building and other fields to denote a ratio or a scale, as in 3:1 (pronounced “three to one”). When a ratio is reduced to a simpler form, such as 10:15 to 2:3, this may be expressed with a double colon as 10:15::2:3; this would be read "10 is to 15 as 2 is to 3". Unicode provides a distinct character U+2236 ∶ ratio for mathematical usage.

The notation |G:H| may also denote the index of a subgroup.

The notation ƒ: X → Y indicates that ƒ is a function with domain X and codomain Y.

The combination with an equal sign, := , is used for definitions.

In mathematical logic, when using set-builder notation for describing the characterizing property of a set, it is used as an alternative to a vertical bar (which is the ISO 31-11 standard), to mean “such that”. Example:

(S is the set of all x in (the real numbers) such that x is strictly greater than 1 and strictly smaller than 3)

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:

Some languages like Haskell use a double colon (::) to indicate type instead.

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