151 Surjective
A surjective mapping is a function that hits every element of its target set that is used to express that a mapping covers its codomain.
Definition (Surjective) A property of a function from a set \(X\) to a set \(Y\), \(F: X \rightarrow Y\), where the following condition applies:
- For every element \(y\) in \(Y\), there exists an element \(x\) in \(X\) such that \(F(x) = y\).
where
- \(X\) is the domain of \(F\).
- \(Y\) is the co-domain of \(F\).
- \(F(X) = Y\) is the statement that the image equals the co-domain.
- \(F(X) = \{y \in Y : y = F(x) \text{ for some } x \in X\}\) is the image of \(X\) under \(F\).
Note:
- \(F(X) = Y\) means the range of the function equals its co-domain.
151.1 Elementary Example
151.1.1 Simple
A surjective map hits every element of the codomain.
\[ F : X \rightarrow Y \]
\[ X = \{ 1,\ 2,\ 3 \},\quad Y = \{ a,\ b \} \]
\[ F(1) = a,\quad F(2) = b,\quad F(3) = a \]
where
- every element of \(Y\) is \(F(x)\) for some \(x \in X\).
- \(F(X) = Y\).