I'm reading about partial ordering, I know by definition it means reflexive, antisymmetric, and transitive. But the examples in my textbook that confused me are of the form:
There is a ordering/relation, and the "less than" of this relation is defined as follow [...] Now by adding equality to this ordering it becomes a partial ordering.
But how can I prove that the ordering/relation is reflexive only given its "less than" condition? e.g.
Say this ordering relation is $R$ and it's denoted by $\prec$. For any $a\in R,$ since $a\not\prec a$ because neither $(a_1, a_2,\dots,a_t)\prec$ itself nor $m\lt n$ (since $m=n$). Then I stuck at this step.
Am I on the correct path proving reflexive from antisymmetric? Or reflexive is defined on the set, so this property don't have to be proved?
another one that similar but I also don't know how to prove: