Im practicing some properties of relations and I cant seem to figure out one particular question. It follows
Consider the relation R on Z+(positive integers) as: For all m,n belonging to Z+, mRn means m|n.
Is R reflexsive, symmetric or transitive?
Provide a complete proof or counterexample for each property.
You may only use the definition of divides
The definition of divides is as follows according to my particular textbook.
let n,d ∈ ℤ+ and d≠0.
n is divisible by d if and only if ∃ k ∈ ℤ such that n = dk
How would I go about solving this? Any and all help is appreciated. Thank you
It's nothing but a matter of exloiting the definition. Since the relation is in words '$m$ divides $n$' it's clear from start that it will be reflexive, transitive but not symmetric.
Indeed $m\mid m$ since $m=1\cdot m$ and so the definition applies
For the transitivity: from $m\mid n$ and $n\mid p$ we have that $n=m\cdot k$, $ p=n\cdot l$. Combining the two we have $$p=n\cdot l=m\cdot k\cdot l\implies m\mid p$$
Symmetry is wrong: $2\mid 4$ but $4\not\mid 2$.