One-One (Injective) Functions

3. One-One (Injective) Functions

Recall from Class XI that a function $f : A \to B$ assigns to every element of $A$ a unique element of $B$. We now classify functions by how they use the codomain $B$.

In an arrow diagram, a one-one function never lets two different starting points land on the same target — every element of the range has exactly one arrow pointing to it.

Method — Proving Injectivity Algebraically

The standard technique assumes $f(x_1) = f(x_2)$ and algebraically forces $x_1 = x_2$.

Illustration. Let $f : \mathbb{R} \to \mathbb{R}$, $f(x) = 5x - 7$. Suppose $f(x_1) = f(x_2)$:

So $f$ is one-one.

Method — Disproving Injectivity by a Counterexample

A single pair of distinct inputs sharing an output is enough to disprove injectivity.

Illustration. Let $f : \mathbb{R} \to \mathbb{R}$, $f(x) = x^2$. Here $f(2) = 4$ and $f(-2) = 4$, so $f(2) = f(-2)$ but $2 \neq -2$. Hence $f$ is not one-one on $\mathbb{R}$.

One-One Functions on Finite Sets …