136 Bijective
A bijective mapping is a function that is both injective and surjective that is used to set up a one-to-one correspondence between two sets.
Definition (Bijective) A property of a function from a set \(X\) to a set \(Y\), \(F: X \rightarrow Y\), where the following conditions apply:
- The function \(F\) is injective
- The function \(F\) is surjective
where
- \(X\) is the domain of \(F\).
- \(Y\) is the co-domain of \(F\)
136.1 Elementary Example
136.1.1 Simple
A bijective map is injective and surjective: a one-to-one correspondence.
\[ F : X \rightarrow Y \]
\[ X = \{ 1,\ 2,\ 3 \},\quad Y = \{ a,\ b,\ c \} \]
\[ F(1) = a,\quad F(2) = b,\quad F(3) = c \]
where
- \(F\) is injective and surjective.
- \(X\) and \(Y\) have the same number of elements.