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.

Comprehend the relationship between unemployment and the PPF.

Grasp the concept of the marginal rate of transformation and how it relates to the PPF slope.

Understand the various ways an agency relationship can be created and recognized under the law.

Identify the differences between an agent and an employee and the legal implications thereof.

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Definitions:

Population

The entire group of individuals or instances about whom we seek to make conclusions.

Statistic

A numerical measurement describing some characteristic of a sample, used to estimate the population parameter.

Sample

A smaller group chosen from the larger population for the purpose of gathering statistical data through measurements, observations, or inquiries.

Inferential Statistics

The branch of statistics that focuses on drawing conclusions and making predictions about populations based on samples.