Injection/Examples
Examples of Injections
$2 x$ Function on Integers is Injective but not Surjective
Let $f: \Z \to \Z$ be the mapping defined on the set of integers as:
- $\forall x \in \Z: \map f x = 2 x$
Then $f$ is an injection, but not a surjection.
$2 x + 1$ Function on Integers is Injective
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 an injection.
$-x$ Function on Integers is Injective
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 an injection.
Square Function on $\N$ is Injective
Let $f: \N \to \N$ be the mapping defined as:
- $\forall n \in \N: \map f n = n^2$
Then $f$ is an injection, but not a surjection.
Cube Function is Injective
Let $f: \R \to \R$ be the real function defined as:
- $\forall x \in \R: \map f x = x^3$
Then $f$ is an injection.
Arbitrary Example
Let $S$ be the set $\set {3, 6}$.
Let $T$ be the set $\set {9, 36, 150$.
Let $f: S \to T$ be the square function:
- $\forall x \in S: \map f x = x^2$
Then $f$ is an injection.
Examples of Mappings which are Not Injections
Arbitrary Mapping on Sets
Let $A = \set {a, b, c, d}$.
Let $B = \set {1, 2, 3, 4, 5}$.
Let $f \subseteq {A \times B}$ be the mapping defined as:
- $f = \set {\tuple {a, 1}, \tuple {b, 4}, \tuple {c, 5}, \tuple {d, 4} }$
Then $f$ is not a injection.
Square Function is Not Injective
Let $f: \R \to \R$ be the real square function:
- $\forall x \in \R: \map f x = x^2$
Then $f$ is not an injection.
$\map f x = \dfrac x 2$ for $x$ Even, $0$ for $x$ Odd is Not Injective
Let $f: \Z \to Z$ be the real function defined 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 not an injection.