Show that the relation R on \mathbb {R} defined as  R = \{(a,b) : a \leq b\}, is reflexive, and transitive but not symmetric.

 

 

 

 
 
 
 
 

Answers (1)

R = \{(a,b) : a \leq b\} where R is real numbers

Reflexive

\\(a,a)\in R,\ \forall a\in R \\ \therefore a \leq a

It is reflexive.

Symmetric:

(a,b)\in R then (b,a)\in R, \ \forall a,b\in R

Let us assume a = 2 \qquad b = 3

        (2,3) \in R \Rightarrow 2 \leq 3 \quad [a\leq b]

                                \Rightarrow (2,3) \in R = LHS

        (3,2) \Rightarrow 3\nleq 2 \qquad [b\leq a] = RHS

                \because LHS \neq RHS

It is not symmetric

Transitive(a,b)\in R, \quad (b,c)\in R \Rightarrow (a,c)\in R

a\leq b \quad -(1) \qquad b\leq c \quad -(2) \Rightarrow a\leq c \quad -(3)

From (1) and (2),  if a\leq b & b\leq c \Rightarrow a \leq b \leq c

                                                                    \therefore a\leq c 

Satisfying condition (3)

It is transitive.

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