Injection/Examples/Non-Injection
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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.