For the given relation for (x , y) ∈R if x ≥ y defined on the set of positive integers only transitive relation holds true.
As given in the question,
Given (x , y) ∈R
To check whether relation reflexive, symmetric, antisymmetric, transitive, and/or a partial order is true or not:
a. Reflexive : If (x, x)∈R for all x ∈R
x ≥ x for all x ∈ R which is not true.
b. Symmetric : If 'x' is related to 'y' then 'y' is also related to 'x' for all (x, y) ∈R.
Here, x ≥ y ⇒ y ≥ x for all (x, y) ∈R , which is not true.
c. Antisymmetric : If 'x' is related to 'y' and 'y' is related to 'x' then x = y for all (x , y)∈ R.
Here, x ≥ y and y ≥ x ⇒ x = y is not true.
Not antisymmetric.
d. Transitive: If 'x' is related to 'y' and 'y' is related to 'z' then 'x' is related to 'z' for all x, y, z ∈R.
Here, x ≥ y , y ≥ z ⇒ x ≥ z for all x, y, z ∈R.
True.
Therefore, for the given relation for (x , y) ∈R if x ≥ y defined on the set of positive integers only transitive relation holds true.
The complete question is:
Determine whether the relation is reflexive, symmetric, antisymmetric, transitive, and/or a partial order.
(x, y)∈R if x ≥ y when defined on the set of positive integers.
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