Example of Relation Help

Question

Example of a relation that is reflexive, not symmetric, not transitive but anti-symmetric. I can't think of an example.

Answer 1

Consider the relation $\sim$ on the real numbers $\mathbb{R}$ given by $$a\sim b\iff a\leq b\leq a+1$$ See if you can prove for yourself that it has the properties listed - if you need help I can say more.

Answer 2

You want your relation $R$ to not be transitive, so there should be $a,b,c$ with $aRb$, $bRc$ but not $aRc$. Well, suppose the underlying set has only those three elements $a,b,c$. What else besides $aRb$ and $bRc$ has to be in the relation? Just $aRa$, $bRb$, $cRc$ to make it reflexive. Check that the relation consisting of $aRb$, $bRc$, $aRa$, $bRb$, $cRc$ satisfies the requirements.

Answer 3

Sit four people at a table and let $xRy$ if $x=y$ or $x$ is seated to the left of $y$.