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Sets and functions
by Samuel S. Watson
Introduction
Subsets
Set Builder Notation
Set Operations
Lists
Functions
Function Properties
Function Operations

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Glossar

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Sets and functionsFunction Properties

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According to the definition, a function may map several elements of the domain to the same element of the codomain, and it may also miss elements in the codomain (in other words, fail to map any domain element to them). However, these behaviors can be undesirable in some situations, and we'll want terminology to refer to whether a given function exhibits them.

Definition
A function f is injective if no two elements in the domain map to the same element in the codomain; in other words if f(a) = f(a') implies a=a'.

A function f is surjective if the range of f is equal to the codomain of f; in other words, if b \in B implies that there exists a\in A with f(a) = b.

A function f is bijective if it is both injective and surjective. This means that for every b\in B, there is exactly one a\in A such that f(a) \in b. If f is bijective, then the inverse of f is the function from B to A that maps b\in B to the element a \in A that satisfies f(a) = b.

Exercise
Identify each of the following functions as injective or not injective, surjective or not surjective, and bijective or not bijective.

  • f:\mathbb{R} \to \mathbb{R}, f(x) = x^2 and
  • f:[0,\infty) \to \mathbb{R}, f(x) = x^2 and
  • f:[0,\infty) \to [0,\infty), f(x) = x^2 and
  • f:\mathbb{R} \to [0,\infty), f(x) = x^2 and

Exercise
For each of the four combinations of injectivity and surjectivity, come up with a real-world function which has that property.

For example, the function from the set of ticket numbers for a commercial airplane flight to the set of passengers on the plane (the one which associates each ticket number with the passenger named on that ticket) is bijective.

Bruno
Bruno Bruno