How can I show that R is a partial ordering on X Let X 5 4

How can I show that R is a partial ordering on X?

Let X = {-5, -4, -3, -2, -1,0, 1, 2, 3,4, 5} For x, y X, set x R y if x^2

Solution

To prove that relation is partial order it must be reflexive, antisymmetric and transitive

Relation to be reflexive if (a,a) belongs to R

Since a = a, hence the relation is reflexive

Relation is said to be antisymmetric if (a,b) belongs to R, then (b,a) must not belong to R

(a,b) belongs to R implies either a^2 < b^2 or a=b

(b,a) implies to R gives that b^2 < a^2, since the first statement says that a^2 < b^2, hence the second statement will be false

Hence the relation is antisymmetric

Relation is said to be transitive if (a,b) and (b,c) belongs to R, then (a,c) must belongs to R

Since a^2 < b^2 and b^2 < c^2, hence we get a^2 < c^2

Therefore, we get (a,c) belongs to R

Hence the relation is a partial order on X