involutsiooni

Involutsiooni is a mathematical concept that describes a transformation of a set onto itself such that applying the transformation twice returns every element to its original position. In other words, if a function f is an involutions, then f(f(x)) = x for all x in the domain of f. This property means that the function is its own inverse.

Examples of involutions are common in various areas of mathematics. In geometry, reflections are involutions. If you reflect a point across a line, reflecting it again across the same line brings it back to its starting position. Similarly, rotations by 180 degrees are also involutions. In set theory, swapping two elements in a set is an involution. If you swap the positions of elements a and b, swapping them again restores their original places.

The concept of involutions is also found in abstract algebra. For instance, in group theory, elements of order 2 within a group are involutions. The set of all involutions of a given set forms a group under composition, where the operation is applying one involution after another. This group is known as the symmetric group on that set, but only considering elements that are their own inverses. The study of involutions helps in understanding the structure of transformations and their properties, particularly in relation to symmetry and self-inverse operations.