Relations on {F,T} are reflexive, irreflexive, symmetric, anti-symmetric and transitive? This is not homework or a test. I just want to better understand when a relation on a set is reflexive, irreflexive, symmetric, anti-symmetric and transitive. < **AND** * How can it be antisymmetric and symmetric? Why is it not just symmetric since the off main-diagonal 0 is mirrored by 0? * Why Transitive? **OR** * Why is it transitive? **IF THEN** * Again why transitive? **IF AND ONLY IF** * Why antisymmetric? * Why transitive? Thank you in advance!

You seem to be struggling with anti0symmetry and Transitivity.

OK, first anti-symmetry.

Anti-symmetric is that for any $a \
eq b$: if $aRb$, then not $bRa$

This is vacuously satisfied for both the AND and the IF AND ONLY IF, since in both cases you simply don't have any $aRb$ with $a$ and $b$ distinct in the first place. So, both are anti-symmetric.

But, they are also symmetric. Like you say: it's mirrored by the diagonal

And yes, it's confusing that something can be symmetrical as well as anti-symmetrical ... but this is what the definitions are, and they are both symmetric and anti-symmetric.

OK, then transitivity:

For transitivity we need:

For any $a,b,c$:L if $aRb$ and $bRc$, then $aRc$

OK, I'll just do the AND, and I'll let you do the others.

Now, for AND, the only time we have $aRb$ and $bRc$ is when $a=b=c=1$. And, we do have $1R1$, i.e. we do have $aRc$ in that case. So, we're good, and the AND is therefore indeed transitive.