Partial ordering on Natural numbers

Question

Preparing for exams, and came across this past year question. Any ideas?

I know that for partial order, it must be reflexive, transitive and anti-symmetric, but how exactly do i show this?

Answer 1

• reflexive: You've done it.

• anti-symmetric: let's see what happens if $x \preceq y$, $y \preceq x$, but $x\neq y$ : we have $3x \le y$ and $3y \le x$, so $9x \le x$, so $x=y=0$, impossible

• transitive: if $x \preceq y$ and $y \preceq z$, and x!=y and y!=z (otherwise the relation is trivial), then $3x \le y$ and $3y \le z$ so $(3x \le) 9x \le z$ , $3x \le z$ so $x \preceq z$