Prove that a relation is antisymmetric

Question

I am a bit confused about proving that a relation is antisymmetric. $x,y \in R$ and $x \sim y$ if $x=2y$ is the given relation. Antisymmetric holds true I think. Could I make this conclusion? Thanks!

consider also $0$, note that $0\sim0$ – Mirko Mar 15 '21 at 03:10 — Mirko, Mar 15 '21 at 03:10

score 0 · Accepted Answer · answered Mar 15 '21 at 03:13

In order to prove that $\sim$ is antisymmetric, you must show that for all $x,y\in R$ it is the case that if $x\sim y$ and $y\sim x$, then $x=y$; looking at one pair of $x$ and $y$ proves nothing. For this relation you can argue that if $x\sim y$, then $x=2y$, and if $y\sim x$, then $y=2x$, so if $x\sim y$ and $y\sim x$, then $x=2y$ and $y=2x$. From that you can infer that $x=4x$, which is true if and only if $x=0$. Thus, $\sim$ is antisymmetric if and only if it is a relation on $\{0\}$. You didn’t tell us what the underlying set is, but I expect that it contains more than just $0$, so … ?

Okay, so its not antisymmetric then. – Mar 15 '21 at 03:15 — , Mar 15 '21 at 03:15
@trevorbates987: Yes, that’s right. – Brian M. Scott Mar 15 '21 at 03:15 — Brian M. Scott, Mar 15 '21 at 03:15

Prove that a relation is antisymmetric

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