Bijection and Inverse element

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Difference between Bijection and Inverse element

Bijection vs. Inverse element

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. In abstract algebra, the idea of an inverse element generalises concepts of a negation (sign reversal) in relation to addition, and a reciprocal in relation to multiplication.

Codomain

In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall.

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Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

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Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.

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Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

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Identity function

Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument.

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If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

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Inverse function

In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function applied to an input gives a result of, then applying its inverse function to gives the result, and vice versa.

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Partial function

In mathematics, a partial function from X to Y (written as or) is a function, for some subset X ′ of X.

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Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

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Symmetric inverse semigroup

In abstract algebra, the set of all partial bijections on a set X (one-to-one partial transformations) forms an inverse semigroup, called the symmetric inverse semigroup (actually a monoid) on X. The conventional notation for the symmetric inverse semigroup on a set X is \mathcal_X or \mathcal_X In general \mathcal_X is not commutative.

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