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- Reflexivity. For all we have because is not a positive integer. The realtion is NOT reflexive.
Symmetry. Let . Then is a positive integer and is negative. Therefore, and the relation is NOT symmetric because the following is always true:
Antisymmetry. Let and . Then is a positive integer and is a positive integer from the definition. Therefore and . There are no such numbers that satisfy and . Therefore, is a contradiction and the relation is antisymmetric because the following is always true:
- Transitivity. Let and . Then for all we have , and therefore . Thus Then is an odd positive integer and is a odd positive integer from the definition. . The difference of two even numbers is even, therefore . The relation is NOT transitive.
Answer. R can be neither an equivalence relation nor a partial ordered set because it is at least not reflexive.