Return to Glossary.

**Formal Definition**

A function $\phi: X\rightarrow Y$ is **one-to-one** if $\phi(x_1) = \phi(x_2)$ only when $x_1=x_2$.

**Informal Definition**

A function $\phi: X\rightarrow Y$ is one-to-one if different inputs map to different outputs.

**Example(s)**

Let $\phi:\mathbb{R}\rightarrow \mathbb{R}$ be defined by $\phi(x)=2x+1$. Then $\phi$ is one-to-one. To see this, suppose $\phi(a)=\phi(b)$. Then

(1)So, we've shown $\phi(a)=\phi(b)$ implies $a=b$, which shows $\phi$ is one-to-one.

**Non-example(s)**

Define $f:2\mathbb{Z}\rightarrow 2\mathbb{Z}$ by $f(a)=a^2$. Then $f$ is not one-to-one, since, for example, $f(-2)=4=f(2)$.

**Additional Comments**

To show that a function $f$ is one-to-one, suppose that $f(a)=f(b)$ and show this implies $a=b$. To show $f$ is not one-to-one, find elements $a\neq b$ such that $f(a)=f(b)$.