I know that a partial order may or may not be a total order. In fact, a total order is a subset of a partial order. But I was thinking about:
Does total order $\implies$ partial order?
In a total order, we have the connexity property ($a \leq b$ or $b \leq a$) (I am considering $\leq$ as $R$ ($aRb$ or $bRa$)). In a partial order, we have reflexivity instead of the connexity of a total order.
If $a \leq b$ or $b \leq a$, then is it true that the reflexive property holds? I am confused here.
Say $a \leq b$. Then $b$ need not be $\leq a$ right? The reflexive property does not hold?
According to their profile, user "conditionalMethod" was last seen on Mathematics Stack Exchange more than two years ago. Therefore, I am posting that user's answer in comments as a community wiki answer:
Yes, you particularize the property $\forall a\forall b (a\leq b \vee b\leq a)$ to the case $a = b$ to get $\forall a(a\leq a \vee a\leq a)$, which is equivalent to $\forall a(a\leq a)$.