Let R be a relation on the set of natural numbers defined by R = {(m, n) | m div
Practice Questions
Q1
Let R be a relation on the set of natural numbers defined by R = {(m, n) | m divides n}. Is R a partial order?
Yes
No
Only reflexive
Only transitive
Questions & Step-by-Step Solutions
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.
