# Get the answer to "How can I prove that a function is..." (2023)

skylsn

2022-06-13

How can I prove that a function is surjective?
I understood by scrolling through the old posts, that if i have a function like this:
$f:\mathbb{R}-\left\{2\right\}\to \mathbb{R}-\left\{5\right\}\mid f\left(x\right)=\frac{5x+1}{x-2}$
If it is given to me something like this: $f:\mathbb{N}×\mathbb{N}\to \mathbb{N}\mid f\left(\left(n,m\right)\right)={2}^{n-1}\left(2m-1\right)$, how can i prove that is surjective? The fact that i have 2 variable is confusing me. Thanks, I hope the question is well asked.

Rebekah Zimmerman

Step 1
You can follow the definition of surjectivity. A function $f:X\to Y$ is said to be surjective iff for each $y\in Y$ there exists $x\in X$ such that $f\left(x\right)=y$. So, if you want to prove a function $f:X\to Y$ is surjective, your proof looks like the following.
Let $y\in Y$ be given. Take $x=\dots \in X$. Then $f\left(x\right)=\cdots =y$. Therefore, f is surjective.
A function with more than one variable can be understood as a function on the Cartesian product of sets. Actually, the set $\mathbb{N}×\mathbb{N}$ is the set of all pairs (m,n) of natural numbers. Therefore, your proof looks like the following.
Step 2
Let $k\in \mathbb{N}$ be given. Take $\left(m,n\right)=\dots \in \mathbb{N}×\mathbb{N}$. Then $f\left(\left(m,n\right)\right)=\cdots =k$. Therefore, f is surjective.
Here, the part "Take $\left(m,n\right)=\dots \in \mathbb{N}×\mathbb{N}$. can be replaced by "Take $m=\dots \in \mathbb{N}$, and take $n=\dots \in \mathbb{N}$, because a pair is determined by its components.

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