Set builder notation for pairs?

Question

I have a set $S$ of integers. I want to select the element pairs $(i,j)$ of it such that $i=2*j$ and order of elements does not matter. How can I show it with the set builder notation ?

$\{\ (i,j)\ |\ i \in S\ \;and\;\ j \in S\; \ and\;i=2*j\}$ Is it correct and are there any ways to achieve this? Thanks in advance.

Answer 1

$\{\ (i,j)\ |\ i \in S\ \;and\;\ j \in S\; \ and\;i=2*j\}$ Is it correct and are there any ways to achieve this? Thanks in advance.

That is okay, although the use of words should be discouraged, and it can be compacted a bit more.   Any of the following should be acceptable: $${\{(i,j)\mid i\in S, j\in S, i=2j\}\\\{(2j,j)\mid j\in S, 2j\in S\}\\\{(i,j) \in S^2 \mid i= 2 j \} \\\{(2j,j)\in S^2\}}$$

Sometimes there is a trade off between compactness and comprehensibilty.   Choose the version that you feel most clearly conveys the intended message.

Answer 2

It's better if you write: $\{(i,j) \in S^2| i= 2*j \} $