Surjection/Examples
Examples of Surjections
Arbitrary Finite Set
Let $S$ and $T$ be sets such that:
| \(\ds S\) | \(=\) | \(\ds \set {a, b, c}\) | ||||||||||||
| \(\ds T\) | \(=\) | \(\ds \set {x, y}\) |
Let $f: S \to T$ be the mapping defined as:
| \(\ds \map f a\) | \(=\) | \(\ds x\) | ||||||||||||
| \(\ds \map f b\) | \(=\) | \(\ds x\) | ||||||||||||
| \(\ds \map f c\) | \(=\) | \(\ds y\) |
Then $f$ is a surjection.
Negative Function on Integers
Let $f: \Z \to \Z$ be the mapping defined on the set of integers as:
- $\forall x \in \Z: \map f x = -x$
Then $f$ is a surjection.
Doubling Function on Reals
Let $f: \R \to \R$ be the real function defined as:
- $\forall x \in \R: \map f x = 2 x$
Then $f$ is a surjection.
Floor Function of $\dfrac {x + 1} 2$ on $\Z$
Let $f: \Z \to \Z$ be the mapping defined on the set of integers as:
- $\forall x \in \Z: \map f x = \floor {\dfrac {x + 1} 2}$
where $\floor {\, \cdot \,}$ denotes the floor function.
Then $f$ is a surjection, but not an injection.
$\map f x = \dfrac x 2$ for $x$ Even, $0$ for $x$ Odd
Let $f: \Z \to \Z$ be the mapping defined on the set of integers as:
- $\forall x \in \Z: \map f x = \begin{cases} \dfrac x 2 & : x \text { even} \\ 0 & : x \text { odd} \end{cases}$
Then $f$ is a surjection.
Real Sine Function to $\closedint 1 1$
Let $I$ denote the closed real interval $\closedint {-1} 1$.
Let $f: \R \to I$ be the mapping defined on the set of real numbers as:
- $\forall x \in \R: \map f x = \sin x$
where $\sin$ denotes the sine function.
Then $f$ is a surjection, but not an injection.
Real Square Function to $\R_{\ge 0}$
Let $f: \R \to \R_{\ge 0}$ be the real square function whose codomain is the set of non-negative reals:
- $\forall x \in \R: \map f x = x^2$
Then $f$ is a surjection.
Examples of Mappings which are Not Surjections
Arbitrary Mapping on Sets
Let $A = \set {a, b, c}$.
Let $B = \set {1, 2, 3}$.
Let $f \subseteq {A \times B}$ be the mapping defined as:
- $f = \set {\tuple {a, 2}, \tuple {b, 1}, \tuple {c, 1} }$
Then $f$ is not a surjection.
Square Function is Not Surjection
Let $f: \R \to \R$ be the real square function:
- $\forall x \in \R: \map f x = x^2$
Then $f$ is not a surjection.
$2 x + 1$ Function on Integers Not Surjection
Let $f: \Z \to \Z$ be the mapping defined on the set of integers as:
- $\forall x \in \Z: \map f x = 2 x + 1$
Then $f$ is not a surjection.