Question

Consider the following relations on {1, 2, 3, 4}.

R₁ = {(2,2), (2,3),(2,4),(3,2),(3,3),(3,4)}

R₂ = {(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)}

R₃ = {2,4),(4,2)}

R₄ = {(1,2),(2,3),(3,4)}

R₅ = {(1,1),(2,2),(3,3),(4,4)}

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 answer.

d)   Which of these relations are transitive? Justify your answers.

Answer 1

$a:those\,\,which\,\,contain\,\,\left( 1,1 \right) ,\left( 2,2 \right) ,\left( 3,3 \right) ,\left( 4,4 \right) ,i.e.R_2,R_5\\b:those\,\,which\,\,contain\,\,\left( a,b \right) \,\,together\,\,with\,\,\left( b,a \right) ,i.e.R_3,R_5\\c:those\,\,for\,\,which\,\left( a,b \right) \in R,\left( b,a \right) \in R\Rightarrow a=b\\R_1-no,\left( 2,3 \right) \in R_1,\left( 3,2 \right) \in R_1\\R_2-no,\left( 1,2 \right) \in R_2,\left( 2,1 \right) \in R_2\\R_3-no,\left( 2,4 \right) \in R_3,\left( 4,2 \right) \in R_3\\R_4-yes, \left( 1,2 \right) \in R_4,\left( 2,1 \right) \notin R_4,\left( 2,3 \right) \in R_4,\left( 3,2 \right) \notin R_4,\left( 3,4 \right) \in R_4,\left( 4,3 \right) \notin R_4\\R_5-yes, no\,\,pairs\,\,\left( a,b \right) \,\,with\,\,a\ne b\\R_4,R_5\\d:those\,\,which\,\,contain\,\,\left( a,c \right) \,\,together\,\,with\,\,\left( a,b \right) ,\left( b,c \right) ,i.e.R_1,R_2,R_5$