Consider the Following Relations on the Set of Positive Integers

question 9

Essay

Consider the following relations on the set of positive integers. $$\begin{aligned}R _ { 1 } & = \{ ( x , y ) \mid x + y > 10 \} \\R _ { 2 } & = \{ ( x , y ) \mid y \text { divides } x \} \\R _ { 3 } & = \{ ( x , y ) \mid \operatorname { gcd } ( x , y ) = 1 \} \\R _ { 4 } & = \{ ( x , y ) \mid x \text { and } y \text { have the same prime divisors } \}\end{aligned}$$
(a) Which of these relations are reflexive? Justify your answers.
(b) Which of these relations are symmetric? Justify your answers.
(c) Which of these relations are antisymmetric? Justify your answers.
(d) Which of these relations are transitive? Justify your answers.

---

Definitions:

Profit Margin

a financial metric used to assess a company's profitability by dividing net income by net sales.

Specialty Products

Goods or services that are perceived as unique or high-value by consumers, often associated with specific buying criteria and loyalty.

Antiques

Items of high value due to their age, rarity, condition, utility, or personal emotional connection, often considered collectible or desirable for their historical significance.

Product Portfolio

A product portfolio is the collection of all products and services offered by a company, categorized and managed to achieve business objectives.