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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.

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What do you mean be the equation of a set? If you want to "plot" these couples, then for each $x$, the second element can take any value in $x+\Bbb Z$. So you will actually draw all lines $y=x+n$, for $n \in \Bbb Z$. These are infinitely many parallel lines (and parallel to the "first diagonal", which is only in the set). [Notice the $y$ I use is not the same as yours, it denotes really the second element of the couple, while $x$ denotes the first]. – Jean-Claude Arbaut Dec 23 '13 at 19:44
Usually we have some equation, like $y=x+2$. Here there is none. That is for me very confusing. And how do I know what $\mathbb{Z}$ is. – depecheSoul Dec 23 '13 at 19:51
$\Bbb Z$ is the set of integers: $\{\dots,-2,-1, 0, 1, 2,\dots\}$ – Jean-Claude Arbaut Dec 23 '13 at 19:54
So it is every integer, not just one value? Now I understand. Thanks. – depecheSoul Dec 23 '13 at 19:58
up vote 2 down vote accepted

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.

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One question. If I set for example x as 1, and I change k. k is infinite, so will I always have x as 1? This is a little bit complicated for me, sorry if it is a stupid question. – depecheSoul Dec 23 '13 at 20:07
@depecheSoul First off, you shouldn't accept an answer if you don't understand it. Answering your comment, what you say is true, but because of the equality $\{(x,x+y)\colon x\in\mathbb{R}\land y\in\mathbb{Z}\}=\bigcup\limits_{k\in\mathbb Z}l_k$, you can be sure that you're not losing any information. And don't apologize for asking questions. – Git Gud Dec 23 '13 at 20:11

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