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Any involution is a bijection.. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x ↦ −x), reciprocation (x ↦ 1/x), and complex conjugation (z ↦ z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the ...
In this example, a self-adjoint morphism is a symmetric relation. The category Cob of cobordisms is a dagger compact category , in particular it possesses a dagger structure. The category Hilb of Hilbert spaces also possesses a dagger structure: Given a bounded linear map f : A → B {\displaystyle f:A\rightarrow B} , the map f † : B → A ...
Involutive negation (unary) can be added as an additional negation to t-norm logics whose residual negation is not itself involutive, that is, if it does not obey the law of double negation . A t-norm logic L {\displaystyle L} expanded with involutive negation is usually denoted by L ∼ {\displaystyle L_{\sim }} and called L {\displaystyle L ...
In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix A n × n {\displaystyle {\mathbf {A}}_{n\times n}} is an involution if and only if A 2 = I , {\displaystyle {\mathbf {A}}^{2}={\mathbf {I}},} where I {\displaystyle {\mathbf {I}}} is the n × n {\displaystyle n\times n ...
As a further example, negation can be defined in terms of NAND and can also be defined in terms of NOR. Algebraically, classical negation corresponds to complementation in a Boolean algebra, and intuitionistic negation to pseudocomplementation in a Heyting algebra. These algebras provide a semantics for classical and intuitionistic logic.
In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve.
It is frequently the case that antiautomorphisms are involutions, i.e. the square of the antiautomorphism is the identity map; these are also called involutive antiautomorphism s. For example, in any group the map that sends x to its inverse x −1 is an involutive antiautomorphism.
As the standard negator is used in the above definition of a t-norm/t-conorm pair, this can be generalized as follows: A De Morgan triplet is a triple (T,⊥,n) such that [1] T is a t-norm; ⊥ is a t-conorm according to the axiomatic definition of t-conorms as mentioned above; n is a strong negator