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.
Read more about this topic: Colon (punctuation)
Famous quotes containing the words mathematics and/or logic:
“The three main medieval points of view regarding universals are designated by historians as realism, conceptualism, and nominalism. Essentially these same three doctrines reappear in twentieth-century surveys of the philosophy of mathematics under the new names logicism, intuitionism, and formalism.”
—Willard Van Orman Quine (b. 1908)
“Somebody who should have been born
is gone.
Yes, woman, such logic will lead
to loss without death. Or say what you meant,
you coward . . . this baby that I bleed.”
—Anne Sexton (1928–1974)