Function
Definition
A function (or mapping) \(f\) from a set \(A\) to a set \(B\) is a relation that assigns to each element of \(A\) exactly one element of \(B\).
\[f: A \to B\]
Formally, \(f \subseteq A \times B\) such that for every \(a \in A\), there exists exactly one \(b \in B\) with \((a, b) \in f\).
Notation
| Symbol | Meaning |
|---|---|
| \(f: A \to B\) | \(f\) is a function from \(A\) to \(B\) |
| \(f(a)\) | The image of \(a\) under \(f\) |
| \(A\) | Domain |
| \(B\) | Codomain |
| \(\{f(a) : a \in A\}\) | Range (or image) |
Properties
Well-definedness
For every \(a \in A\), there exists exactly one \(b \in B\) such that \(f(a) = b\).
Domain and Codomain
The domain is the set of all inputs. The codomain is the set of all possible outputs. The range is the set of actual outputs.
Examples
Example 1: Identity Function
\[id_A: A \to A, \quad id_A(a) = a\]
Example 2: Square Function
\[f: \mathbb{R} \to \mathbb{R}, \quad f(x) = x^2\]
Range: \([0, \infty)\)
Example 3: Constant Function
\[g: A \to B, \quad g(a) = b_0 \text{ for all } a \in A\]
Related
- Cartesian Product — The domain of a function is a subset of this
- Relation — A function is a special kind of relation
- Set — Functions map between sets