INVERSE

inverse, reverse

(adjective) reversed (turned backward) in order or nature or effect

inverse

(adjective) opposite in nature or effect or relation to another quantity; “a term is in inverse proportion to another term if it increases (or decreases) as the other decreases (or increases)”

inverse, opposite

(noun) something inverted in sequence or character or effect; “when the direct approach failed he tried the inverse”

Source: WordNet® 3.1

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Etymology

Adjective

inverse (not comparable)

Opposite in effect, nature or order.

Reverse, opposite in order.

(botany) Inverted; having a position or mode of attachment the reverse of that which is usual.

(mathematics) Having the properties of an inverse; said with reference to any two operations, which, when both are performed in succession upon any quantity, reproduce that quantity.

(geometry) That has the property of being an inverse (the result of a circle inversion of a given point or geometrical figure); that is constructed by circle inversion.

(category theory, of a category) Whose every element has an inverse (morphism which is both a left inverse and a right inverse).

Noun

inverse (plural inverses)

An inverted state: a state in which something has been turned (properly) upside down or (loosely) inside out or backwards.

The result of an inversion, particularly

The reverse of any procedure or process.

(mathematics) A ratio etc. in which the antecedents and consequents are switched.

(geometry) The result of a circle inversion; the set of all such points; the curve described by such a set.

(logic) The non-truth-preserving proposition constructed by negating both the premise and conclusion of an initially given proposition.

"Anything that isn't a dog doesn't go to heaven" is the inverse of "All dogs go to heaven." More generally, $\backslash (\backslash lnot\backslash mathsf\{p\}\backslash to\backslash lnot\backslash mathsf\{q\}\backslash )$ is the inverse of $\backslash (\backslash mathsf\{p\}\backslash to\backslash mathsf\{q\}\backslash )$ and is equivalent to the converse proposition $\backslash (\backslash mathsf\{q\}\backslash to\backslash mathsf\{p\}\backslash )$.

(mathematics) A second element which negates a first; in a binary operation, the element for which the binary operation—when applied to both it and an initially given element—yields the operation's identity element, specifically:

(addition) The negative of a given number.

The additive inverse of $\backslash (x\backslash )$ is $\backslash (-x\backslash )$, as $\backslash (x-x=0\backslash )$, as $\backslash (0\backslash )$ is the additive identity element.

(multiplication) One divided by a given number.

The multiplicative inverse of $\backslash (x\backslash )$ is $\backslash (x^\{-1\}\backslash )$, as $\backslash (x\; \backslash times\; x^\{-1\}=1\backslash )$, as $\backslash (1\backslash )$ the multiplicative identity element.

(functions) A second function which, when combined with the initially given function, yields as its output any term inputted into the first function.

The compositional inverse of a function $\backslash (f\backslash )$ is $\backslash (f^\{-1\}\backslash )$, as $\backslash (f\backslash \; f^\{-1\}=\backslash mathit\{I\}\backslash )$, as $\backslash (\backslash mathit\{I\}\backslash )$ is the identity function. That is, $\backslash (\backslash forall\; x,f(f^\{-1\}(x))=\backslash mathit\{I\}(x)=x\backslash )$.

(category theory) A morphism which is both a left inverse and a right inverse.

(card games) The winning of the coup in a game of rouge et noir by a card of a color different from that first dealt; the area of the table reserved for bets upon such an outcome.

(linguistics, Kiowa-Tanoan) A grammatical number marking that indicates the opposite grammatical number (or numbers) of the default number specification of noun class.

Synonyms

• (addition): additive inverse

• (multiplication): multiplicative inverse

• (composition): compositional inverse

• (geometry): inverse point, inverse curve

Verb

inverse (third-person singular simple present inverses, present participle inversing, simple past and past participle inversed)

(surveying) To compute the bearing and distance between two points.

Antonyms

• compute (a point).

Anagrams

• Severin, enviers, inserve, venires, versine

Source: Wiktionary

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In*verse", a. Etym: [L. inversus, p. p. of invertere: cf. F. inverse. See Invert.]

1. Opposite in order, relation, or effect; reversed; inverted; reciprocal; -- opposed to direct.

2. (Bot.)

Definition: Inverted; having a position or mode of attachment the reverse of that which is usual.

3. (Math.)

Definition: Opposite in nature and effect; -- said with reference to any two operations, which, when both are performed in succession upon any quantity, reproduce that quantity; as, multiplication is the inverse operation to division. The symbol of an inverse operation is the symbol of the direct operation with -1 as an index. Thus sin-1 x means the arc whose sine is x. Inverse figures (Geom.), two figures, such that each point of either figure is inverse to a corresponding point in the order figure.

– Inverse points (Geom.), two points lying on a line drawn from the center of a fixed circle or sphere, and so related that the product of their distances from the center of the circle or sphere is equal to the square of the radius.

– Inverse, or Reciprocal, ratio (Math.), the ratio of the reciprocals of two quantities.

– Inverse, or Reciprocal, proportion, an equality between a direct ratio and a reciprocal ratio; thus, 4 : 2 : : , or 4 : 2 : : 3 : 6, inversely.

In"verse, n.

Definition: That which is inverse. Thus the course of human study is the inverse of the course of things in nature. Tatham.

Source: Webster’s Unabridged Dictionary 1913 Edition

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