Determine whether the relation N times N lessthanorequalto w

Determine whether the relation (N times N, lessthanorequalto where (a, b) lessthanorequalto (c, d) if and only if a lessthanorequalto c and b greaterthanorequalto d is a partial order.

Solution

Here by rule, a relation is a partial order, if it is transitive and antisymmetric.

Now let we have three ordered pairs (a,b),(c,d) and (e,f)

then if (a,b) < (c,d), i.e. a<c, and b> d

and (c,d) < (e,f) i.e. c<e and d >f

then clearly a<c<e and b > d>f

or a<e and b>f i.e. (a,b) <(e.f)

Thus this relation is surely transitive.

Now for antisymmetry, we show that If (a,b) < (c,d) then (c,d) <(a,b) must not hold that is true in this case as if a<c then c can not be smaller than a.

Thus given relation is as both transitive and antisymmetric, is a partial order.