Functions - 2
Injective (one-to-one) Functions
Let $f:A\rightarrow B$ be a function. We call $f$ and injective, or one-to-one, function if:
\[f(a_1)=f(a_2)\Rightarrow a_1 = a_2 \text{ for all } a_1,a_2\in A\]This is logically equivalent to $a_1\neq a_2 \Rightarrow f(a_1) \neq f(a_2)$ and so injective functions never repeat values. In other words:
Different inputs give different outputs.
Example 1
$f:\mathbb{Z}\rightarrow \mathbb{Z}$ given by $f(x)=x^2$ is not injective.
$h:\mathbb{Z}\rightarrow \mathbb{Z}$ given by $h(x)=2x$ is injective.
Example 2
To prove that a function is not injective you can give an individual example of a double mapping.
Take the following question foe the opposite:
$h:\mathbb{Z}\rightarrow \mathbb{Z}$ given by $h(x)=2x$ is injective.
Proof
Suppose for a proof by contradiction that there exist $a_1,a_2$ such that $h(a_1=h(a_2)$ and $a_1\neq a_2$.
$2\times a_1 = 2a_2 \Rightarrow a_1 = a_2$, a contradiction.
Surjective (or onto) Functions
$f:A\rightarrow B$ is surjective, or onto, if the range of $f$ coincides with the co-domain $f$. This means that for every $b\in B$ there exists an $a\in A$ with $b=f(a)$.
Examples
$h:\mathbb{Z}\rightarrow \mathbb{Z}$ given by $h(x)=2x$ is not surjective.
This is because you get every even values out as an answer.
$h’:\mathbb{Q}\rightarrow \mathbb{Q}$ given by $h’(x)=2x$ is surjective.
This is as you can use rational numbers to make any other number when doubled.
Question
Classify $f:\{a,b,c\}\rightarrow\{1,2,3\}$ given by:
graph LR
subgraph x
a
b
c
end
subgraph fx
1
2
3
end
a --> 1
b --> 1
c --> 3
2
- It is a function.
- Not injective, $f(a)=f(b)=1$
- Not subjective as no $x$ maps with $f(x)=2$.
Bijections
We call $f$ bijective if $f$ is both injective and surjective.
Examples
$f:\mathbb{Q}\rightarrow \mathbb{Q}$ given by $f(x)=2x$ is bijective.
Inverse Functions
If $f$ is a bijection from a set $X$ to a set $Y$, then there is a function $f^{-1}$ from $Y$ to $X$ that undoes the action of $f$; that is, it sends each element of $Y$ back to the element of $X$ that it came from. This function is called the inverse function for $f$.
Then $f(a)=b$ if, and only if, $f^{-1}(b)=a$
Example
$k:\mathbb{R}\rightarrow \mathbb{R}$ given by $k(x)=4x+3$ is invertible and $k^{-1}(y)=\frac{1}{4}(y-3)$.
$y=4x+3$. So $4x+3=y$, $4x=y-3$, $x = \frac{y-3}{4}$
This proves the statement by giving the same value.