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In mathematics, a binary relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c. Every partial order and every equivalence relation is transitive. For example, less than and equality among real numbers are both transitive: If a < b and b < c then a < c; and if x ...
Throughout this article, capital letters (such as ,,,,, and ) will denote sets.On the left hand side of an identity, typically, will be the leftmost set, will be the middle set, and
The manipulations of the Rubik's Cube form the Rubik's Cube group.. In mathematics, a group is a set with an operation that satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.
The Frobenius group of affine transformations of F 5 (maps + where a ≠ 0) has order 20 = (5 − 1) · 5 and acts on the field with 5 elements, hence is a subgroup of S 5. (Indeed, it is the normalizer of a Sylow 5-group mentioned above, thought of as the order-5 group of translations of F 5 .)
An involution is a function f : X → X that, when applied twice, brings one back to the starting point. In mathematics, an involution, involutory function, or self-inverse function [1] is a function f that is its own inverse, f(f(x)) = x. for all x in the domain of f. [2] Equivalently, applying f twice produces the original value.
Since bijections have inverses, so do permutations, and the inverse σ −1 of σ is again a permutation. Explicitly, whenever σ(x)=y one also has σ −1 (y)=x. In two-line notation the inverse can be obtained by interchanging the two lines (and sorting the columns if one wishes the first line to be in a given order). For instance
The identity function on any partially ordered set is always an order automorphism.; Negation is an order isomorphism from (,) to (,) (where is the set of real numbers and denotes the usual numerical comparison), since −x ≥ −y if and only if x ≤ y.
In a vector space, the additive inverse −v (often called the opposite vector of v) has the same magnitude as v and but the opposite direction. [11] In modular arithmetic, the modular additive inverse of x is the number a such that a + x ≡ 0 (mod n) and always exists. For example, the inverse of 3 modulo 11 is 8, as 3 + 8 ≡ 0 (mod 11). [12]