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.

Question

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.

Quizplus · Accepted Answer

The Answer of 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.

Consider the Following Relations on the Set of Positive Integers