Define a relation R on Z, the set of all integers as follows: For every ,
Is R a partial order relation? Prove or give a counterexample.
Whether is a partial order relation or not.
For all every prime factor of is a prime factor of .
A relation that is reflexive, antisymmetric, and transitive is called a partial order.
To verify anti-symmetricity:
The relation is not anti-symmetric.
Observe that, because every prime factor of is a prime factor of .
Also that because every prime factor of is a prime factor of .
So, the relation is not anti-symmetric.
To verify transitivity:
Suppose that such that
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