Consider the Following Relations On $\{ 1,2,3 \}$ (A) Which of These Relations Are Reflexive? Justify Your Answers

question 4

Essay

Consider the following relations on $\{ 1,2,3 \}$ .
$$\begin{array} { l } R _ { 1 } = \{ ( 1,1 ) , ( 1,3 ) , ( 2,2 ) , ( 3,1 ) \} \\R _ { 2 } = \{ ( 1,1 ) , ( 2,2 ) , ( 3,1 ) , ( 3,3 ) \} \\R _ { 3 } = \{ ( 1,2 ) , ( 2,1 ) , ( 3,3 ) \} \\R _ { 4 } = \{ ( 1,3 ) , ( 2,3 ) \}\end{array}$$
(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.

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Definitions:

Manufacturing Overhead

Expenses related to the manufacturing process that are not directly attributed to individual units produced, such as maintenance and factory supervision.

Machine-Hours

A measure of the time a machine is operated, used as a basis for allocating manufacturing overhead in some costing systems.

Predetermined Overhead Rate

A calculated rate designed for allocating manufacturing overhead expenses to specific production units, based on a selected activity base.

Manufacturing Overhead

All indirect costs associated with manufacturing, excluding direct materials and direct labor costs. These costs include indirect materials, indirect labor, and other expenses.