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

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.