If R is a relation defined on the set Z by a R b if a-b is a non negative even integer. Determine if R define a partial order and total order.

The relation R needs to be reflexive, antisymmetric, and transitive to be a partial order. All these conditions are obeyed by R as stated above. In fact, the popular notation to represent a partial order is derived from precisely the same thing (a-b being a non-negative integer).

Reflexive: aRa for all an in Z, since a-a=0, a non-negative integer, for any a.

The relation R needs to be reflexive, antisymmetric, and transitive to be a partial order. All these conditions are obeyed by R as stated above. In fact, the popular notation to represent a partial order is derived from precisely the same thing (a-b being a non-negative integer).

Antisymmetric: If $aR_b$ and $bR_a$, ie., if $a-b≥0$ and $b-a≥0$, then $a≥b$ and $b≥a$, which is only possible if $a=b$.

Transitive: If $aR_b$ and $bR_c$, then $a-b≥0$ and $b-c≥0$, which gives $a-c≥0$ and hence aRc.R is also a total order, because for any given a or b in Z, either aRb or bRa must be true (This is because either $a≤b$ or $b≤a$ ).