142 Injective
An injective mapping is a function that maps distinct elements to distinct values that is used to express uniqueness of solutions of mapping equations.
Definition (Injective) A property of a function from a set \(X\) to a set \(Y\), \(F: X \rightarrow Y\), where the following condition applies:
- For all elements \(x_1\) and \(x_2\) in \(X\), if \(F(x_1) = F(x_2)\), then \(x_1 = x_2\).
where
- \(F\) is a function from \(X\) to \(Y\).
- \(X\) is the domain of \(F\).
- \(Y\) is the co-domain of \(F\).
- \(x_1, x_2\) are elements of \(X\).
Note:
- the condition is logically equivalent to saying that if \(x_1 \neq x_2\), then \(F(x_1) \neq F(x_2)\).
- for an injective function, each element of the co-domain is the image of at most one element of the domain.
142.1 Elementary Example
142.1.1 Simple
An injective map maps distinct inputs to distinct outputs.
\[ F : X \rightarrow Y \]
\[ X = \{ 1,\ 2,\ 3 \},\quad Y = \{ a,\ b,\ c,\ d \} \]
\[ F(1) = a,\quad F(2) = b,\quad F(3) = d \]
where
- \(F\) is injective because all three values are different.
- \(X\) is the domain and \(Y\) is the codomain.