Invariant (mathematics)

In mathematics, an invariant is a property, held by a class of mathematical objects, which remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases 'invariant under' and 'invariant to' a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. A simple example of invariance is expressed in our ability to count. For a finite set of objects of any kind, there is a number to which we always arrive, regardless of the order in which we count the objects in the set. The quantity—a cardinal number—is associated with the set, and is invariant under the process of counting.A subset S of the domain U of a mapping T: U → U is an invariant set under the mapping when x ∈ S ⇒ T ( x ) ∈ S . {displaystyle xin SRightarrow T(x)in S.}   Note that the elements of S are not fixed, but rather the set S is fixed in the power set of U. (Some authors use the terminology setwise invariant vs. pointwise invariant to distinguish between these cases.)For example, a circle is an invariant subset of the plane under a rotation about the circle’s center. Further, a conical surface is invariant as a set under a homothety of space.The notion of invariance is formalized in three different ways in mathematics: via group actions, presentations, and deformation.In computer science, one can encounter invariants that can be relied upon to be true during the execution of a program, or during some portion of it. It is a logical assertion that is always held to be true during a certain phase of execution. For example, a loop invariant is a condition that is true at the beginning and end of every execution of a loop.