A partial order P is defined on the set of natural numbers as follows. Here ÷ denotes integer division.

i. (0, 0) ∈ P
i...

Question

A partial order P is defined on the set of natural numbers as follows. Here ÷ denotes integer division.

i. (0, 0) ∈ P
ii. (a, b) ∈ P if and only if a ÷ 10 = b ÷ 10 and a mod 3 ≤ b mod 3

Consider the following ordered pairs:
i. (100, 22)
ii. (22, 10)
iii. (148, 26)
iv. (10, 130)

Which of these ordered pairs of natural numbers are contained in P?

A. (i) and (ii)
B. (ii) and (iii)
C. (i) and (iii)
D. (iii) and (iv)

Accepted Answer

D. (iii) and (iv) The partial order P groups natural numbers by their integer-quotient when divided by 10 (i.e., the 'decade group') and orders them within a group by their remainder mod 3. Two numbers are related (a,b) ∈ P exactly when they fall in the same decade-group AND a's remainder mod 3 does not exceed b's remainder mod 3. We check each pair against both conditions.