Determine the set $A$ = {$m\in Z|mR52$} and give its cardinality $|A|$.

Question

Given a relation R on $Z^+$ defined as: $mRn$ if and only if $m|n$, I need to determine the set $A$ = {$m\in Z|mR52$} and give its cardinality $|A|$.

I know that $mR52$ = $m|52$ and that $52 = mk$ for some integer k and, I believe that this will consist of the pairs: (-1, -52), (-2,-26), (-4,-13), (1, 52), (2,26), (4,13) with the cardinality consisting of (1, 52), (2,26), (4,13). But, I am not sure how to write this properly.

Any help is appreciated.

Thanks,

Tony

Answer 1

First note that according to the first line, our relation is defined on $\mathbb{Z}^+$. This is the set of positive integers. So we will not have to worry about negative numbers.

The set $A$ consists of the positive integers $m$ such that $m\text{R}52$. So we want to find the positive integers $m$ that divide $52$. These are $1$, $2$, $4$, $13$, $26$, and $52$.

If you like, you can write $A=\{1,2,4,13,26,52\}$. The cardinality $|A|$ of $A$ is $6$.