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Consider the set A = {0, 1, 2, 3, 4, 5, 6}. The relation $ xRy \iff 2 | (x - y) $. What is relation R?

My answer is R = {(2,0), (4,0), (6,0), (3,1), (5,1), (4,2), (6,2), (6,3), (0,2), (0,4), (0,6), (1,3), (1,5), (2,4), (2,6), (3,5)}.

2 Questions:

  1. Is that correct?
  2. What about pairs like (x, x) like {(0,0), (1,1), ... , (5,5), (6,6) }

Sorry if this is like a simple question but I have just started on relations...and the solutions sheet that I was given wasn't filled.

Any comments/answers is very much appreciated! Thanks in advance!

terahertz
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  • Yes, these are also pairs in the relation. $x-x=0$ is divisible by $2$. – Mark Oct 21 '19 at 14:09
  • In case it is not obvious why $0$ is divisible by $2$, remember how divisibility is defined. We say that $a$ is divisible by $b$ iff there exists some integer $k$ such that $a=bk$. Indeed, letting $a=0,b=2$ we can find such a $k$ such that $0=2k$, that $k$ being $0$. – JMoravitz Oct 21 '19 at 14:11
  • This particular relation in your question is an important one. In more common everyday words, you have $x$ is related to $y$ "If they have the same parity", i.e. if they are both even or if they are both odd. Compare this to the relation where it was $10\mid (x-y)$ which would in more common words be $x$ and $y$ are both related "if they both end in the same digit (in their base 10 representation)." – JMoravitz Oct 21 '19 at 14:13
  • I think $(6,3)$ is supposed to be $(6,4)$ and a few pairs seem to be missing. If $(x,y)\in R$, then also $(y,x)\in R$, since if $2\mid\alpha$ then also $2\mid -\alpha$. – HSN Oct 21 '19 at 14:19

1 Answers1

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R = { (x,y) : x and y are both even or x and y are both odd }
Exercise. Prove R is an equivalence relation.

William Elliot
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