Let R be a relation on the set of natural numbers defined by R = {(m, n) | m div

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Question: Let R be a relation on the set of natural numbers defined by R = {(m, n) | m divides n}. Is R a partial order?

Options:

Correct Answer: Yes

Solution:

R is reflexive, antisymmetric, and transitive, thus it is a partial order.

Let R be a relation on the set of natural numbers defined by R = {(m, n) | m divides n}. Is R a partial order?

Let R be a relation on the set of natural numbers defined by R = {(m, n) | m divides n}. Is R a partial order?

Step 1: Understand what a relation is. A relation R on a set is a way to compare elements of that set. In this case, R compares natural numbers based on whether one number divides another.
Step 2: Recall the definition of a partial order. A relation is a partial order if it is reflexive, antisymmetric, and transitive.
Step 3: Check if R is reflexive. A relation is reflexive if every element is related to itself. For any natural number m, m divides m (since m/m = 1). Therefore, R is reflexive.
Step 4: Check if R is antisymmetric. A relation is antisymmetric if, for any m and n, if m divides n and n divides m, then m must equal n. If m divides n and n divides m, it means they are the same number. Thus, R is antisymmetric.
Step 5: Check if R is transitive. A relation is transitive if, whenever m divides n and n divides p, then m must divide p. If m divides n and n divides p, then m divides p (because of the properties of division). Therefore, R is transitive.
Step 6: Since R is reflexive, antisymmetric, and transitive, we conclude that R is a partial order.

Partial Order – A relation that is reflexive, antisymmetric, and transitive.
Reflexivity – For all elements a in the set, (a, a) is in the relation.
Antisymmetry – For all elements a and b in the set, if (a, b) and (b, a) are in the relation, then a must equal b.
Transitivity – For all elements a, b, and c in the set, if (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation.
Divisibility – A relation where one number can be divided by another without leaving a remainder.