Consider the Following Relations on the Set of Positive Integers

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.