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Section 8.2 Functions

Subsection 8.2.1 Functions

Definition 8.2.1.

Let \(X, Y\) be non-empty sets. A function (or map, or mapping), \(f\text{,}\) from the domain \(X\) (or \(D(f)\)) to the codomain \(Y\text{,}\) \(f: X \rightarrow Y\text{,}\) is a relation between \(X\) and \(Y\) such that,

  1. All elements in the domain map to some element in the codomain. For all \(x \in X\text{,}\) there exists \(y \in Y\) such that \((x,y) \in f\text{.}\)

  2. The function is well-defined. If \((x,y_1) \in f\) and \((x,y_2) \in f\text{,}\) then \(y_1 = y_2\text{.}\)

If \((x,y) \in f\text{,}\) we denote this by \(y = f(x)\text{.}\) Then, these requirements are,

  1. For all \(x \in X\text{,}\) there exists \(y \in Y\) such that \(f(x) = y\text{.}\)

  2. If \(f(x) = y_1\) and \(f(x) = y_2\text{,}\) then \(y_1 = y_2\text{.}\) Equivalently, if \(x_1 = x_2\text{,}\) then \(f(x_1) = f(x_2)\text{.}\)

  • If $f(x) = y$, then we say "$y$ is the image of $x$", or "$x$ maps to $y$".

  • The range of \(f\text{,}\) sometimes denoted by \(R(f)\text{,}\) is the set of all outputs of \(f\text{.}\) In other words,

    \begin{equation*} R(f) = \set{f(x): x \in X} = \set{y \in Y: \exists x \in X \text{ such that } f(x) = y} \end{equation*}
    In general, \(R(f) \neq Y\text{.}\)

Definition 8.2.2.

Functions \(f\) and \(g\) are equal if and only if for all \(x_1 \in X\text{,}\) \(f(x_1) = f(x_2)\text{.}\)

Let \(f = \set{(x,y) \in \mathbb{Z}^2: 3x + y = 0}\text{.}\) Let \(x \in \mathbb{Z}\) such that \(y = -3x\text{.}\) Then, \(3x + (-3x) = 0\) satisfies the condition. Also, if \((x,y) \in f\) and \((x,z) \in f\text{,}\) then \(3x + y = 0\) and \(3x + z = 0\text{.}\) Thus, \(y = z\text{.}\)

Let \(X\) be a non-empty set. The identity function on \(X\) is \(Id_{X}: X \rightarrow X\text{,}\) \(Id_{X}(x) = x\text{.}\)

Operations like addition, subtraction, multiplication, and division can be thought of as functions. For example, addition can be thought of as the function,

\begin{align*} + : \mathbb{R} \times \mathbb{R} \amp \longrightarrow \mathbb{R}\\ (a,b) \amp \longmapsto a + b \end{align*}

In function notation, this is written as \(+((a,b)) = a + b\text{.}\) For example, \(+((2,3)) = 5\text{.}\)

Subsection 8.2.2 Examples of Non-Functions

The \(f(\frac{a}{b}) = a + b\) is not a function on the rational numbers. This is because for example, \(f(\frac{1}{2}) = 1 + 2 = 3\) but also \(f(\frac{1}{2}) = f(\frac{2}{4}) = 2 + 4 = 6 \neq 3\text{.}\)

The output of the function must not depend on the representation of the input element, and only depend on the input element itself.

Let \(f = \set{(x,y) \in \mathbb{Z}^2: 3x + 2y = 0}\text{.}\) Then, \(f\) is not a function, as for \(1 \in \mathbb{Z}\text{,}\) \(3 + 2y = 0\text{,}\) but \(y = -3/2 \notin \mathbb{Z}\text{.}\) Thus, \((1,y) \notin f\text{.}\)