How to sketch a set

Question

For assingment I need to sketch the following set:

$$ \{(x,x+y)|x\in\mathbb{R}, y\in\mathbb{Z}\} $$

I am a little bit confussed, because I am not sure how to set the equation of this set.

Please Help. Thanks.

Answer 1

For each $k\in \mathbb Z$, set $l_k:=\{(x,x+k)\colon x\in \mathbb R\}$. The graph of this set is the graph of the line $y=x+k$, (this $y$ doesn't represent the same $y$ as the one in $\{(x,x+y)\colon x\in\mathbb{R}\land y\in\mathbb{Z}\}$).

Note that $\{(x,x+y)\colon x\in\mathbb{R}\land y\in\mathbb{Z}\}=\bigcup\limits_{k\in\mathbb Z}l_k$, so the graph of $\{(x,x+y)\colon x\in\mathbb{R}\land y\in\mathbb{Z}\}$ is a graph that contains all the graphs of the lines $y=x+k$ and it contains nothing else. Thus the graph of $\{(x,x+y)\colon x\in\mathbb{R}\land y\in\mathbb{Z}\}$ is all the lines with slope $1$ and integer constant term.