Is $f(x, y) = 5x - 4y$ injective or surjective?

Question

Define $f : \Bbb Z\times\Bbb Z \to\Bbb Z$ by $f(x, y) = 5x - 4y$.

Is $f$ injective or surjective?

How would I go about proving this?

Thanks

Answer 1

HINT

• A function is injective if it maps different inputs for different values. Can you find $x,y,v,w,a$ such that $f(x,y) = a = f(v,w)$ with $(x,y) \ne (v,w)$?
• A function is surjective if it completely covers the set it maps to. Let $a \in \mathbb{Z}$. Can you find such $x,y \in \mathbb{Z}$ so that $f(x,y)=a$?

Answer 2

Surjective: Fix a $ z \in \mathbb{Z}$ and try to find an $x=(x1,x2) \in \mathbb{Z}\times\mathbb{Z}$ with $f(x1,x2)=z$. For Injectivity, show that for $(x1,x2) \neq (y1,y2) \in \mathbb{Z}\times\mathbb{Z}\implies f(x1,x2) \neq f(y1,y2)$.

Answer 3

Hints:

$$5x-4y=5a-4b\iff 5(x-a)=4(y-b)\implies\begin{cases}x=a\pmod 4\\y=b\pmod 5\end{cases}$$

For example: $\;5\cdot3-4\cdot7=5\cdot7-4\cdot12\;$

Also, since $\;gcd(4,5)=1\;$ , there exist $\;m,n\in\Bbb Z\;$ s.t. $\;5m+4n=1\;$ , so for any $\;t\in\Bbb Z\;$ ...