Proof of reflexive, symmetric and transitive when m | n

Question

I'm trying to write proofs or counterexample using the definition of divides only for each operation (reflexive, symmetric or transitive) in the following: The relation V on Z+ for all m and n in Z+, mVn <-> m | n.

I'm familiar with the set operations as:

V is reflexive if for all x in A, xVx.

V is symmetric if for all x, y in A, if xVy, then yVx.

V is transitive if for all x, y and z in A, if xVy and yVz, then xVz.

I'm just not sure how to approach divisibility her. Any help would be appreciated!

Answer 1

You have reflexive property because every positive integer divides itself.

You do not have symmetry because for example $5|10$ but $10\not | 5$

You do have transitive property because if $x|y$ and $y|z$ then $x|z$

You can fill in the details of the proof for each part.

This relation is not an equivalence relation.