One To One

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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)
\begin{eqnarray} 2a+1&=&2b+1\\ 2a&=&2b\\ a&=&b \end{eqnarray}

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 $f(a)=f(b)$.

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