Is lprN a linear ordering on N For x y N define x prN y if f

Is lprN a linear ordering on N?

For x, y N, define x |_prN y if, for some z N, z notequal 0, z notequal 1, z ? x = y. We say x is a proper divisor of y. Is |_prN a linear ordering on N?

Solution

A Linear ordering means that a partial ordering inwhich any two elements are comparable.

First we verify that partial order[reflexive, antisymmetric, transtive]

x | x whisch imply that | is reflexive

If x | y and y | x, then we get x=y. therefore | is antisymmetric.

If x | y, and y | z, then x | z. therefore | is transitive.

But any two elements in N doesn\'t need to be comparable [for example, if we take 2 and 3 in N, 2 doesn\'t divide 3 and 3 doesn\'t divide 2.]

Therefore given relation proper divisor | is only partial ordering but not Linear ordering.