Mathematics of Discrete Structures for Computer Science by Gordon J. Pace

Author:Gordon J. Pace
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

We write r:X→Y to express that r:X↔Y and that r is functional.

An injective function is also called a one-to-one function.  ■

Since a function f relates every object in the domain to, at most, one object in the range, we know that, using relation application: f〈〈{x}〉〉 is equal to either {y} or ∅. When dealing with functions, we will be using the more familiar notation f(x) (where x is an element of the domain of f) to represent the unique value y to which x is related. Note that, if f were not functional due to some value x in the domain having more than one outgoing arrow, writing f(x) would be ambiguous and is thus not allowed.

Both injectivity and functionality constrain the number of arrows to be at most one—the former constrains the arrows on the range, while the latter on the domain. If we were to switch the direction of the arrows, we would thus go from an injective relation to a functional one, and vice versa:

Theorem 6.9

A relation r:X↔Y is injective if and only if r −1 is functional. Conversely, r:X↔Y is functional if and only if r −1 is injective.

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