Injection (One-to-One Function)
Definition
A function \(f: A \to B\) is an injection (or one-to-one) if:
\[\forall a_1, a_2 \in A, f(a_1) = f(a_2) \implies a_1 = a_2\]
Equivalently: distinct inputs map to distinct outputs.
Examples
Example 1: \(f(x) = 2x\) on \(\mathbb{R}\)
\[f(x_1) = f(x_2) \implies 2x_1 = 2x_2 \implies x_1 = x_2\]
This is an injection.
Example 2: \(f(x) = x^2\) on \(\mathbb{R}\)
\[f(1) = f(-1) = 1\]
Not an injection (unless domain is restricted to \([0, \infty)\)).
Example 3: Identity Function
\(id_A: A \to A\) is always an injection.
Composition of Injections
If \(f: A \to B\) and \(g: B \to C\) are injections, then \(g \circ f: A \to C\) is an injection.
Related
- Function — The general concept
- Surjection — Every output is hit (dual concept)
- Bijection — Both injection and surjection
- Direct Proof — Proving a function is injective