Let x, y ∈ I and suppose that a relation R on I is defined by x R y if and only if x ≤ y then

Question

Let x, y ∈ I and suppose that a relation R on I is defined by x R y if and only if x ≤ y then (A) R is reflexive but not symmetric (B) R is an equivalence relation (C) R is neither reflexive nor symmetric (D) R is symmetric but not transitive

Tardigrade Admin · Accepted Answer

Since x ≤ x for all x ∈ I so R is reflexive but is not symmetric as (1,2) ∈ R and (2,1) ∉ R. Also R is transitive as x ≤ y, y ≤ z ⇒ x ≤ z.

Q. Let $x, y \in I$ and suppose that a relation $R$ on $I$ is defined by $x R y$ if and only if $x \leq y$ then

$R$ is reflexive but not symmetric

$R$ is an equivalence relation

$R$ is neither reflexive nor symmetric

$R$ is symmetric but not transitive

Since $x \leq x$ for all $x \in I$ so $R$ is reflexive but is not symmetric as $(1, 2) \in R$ and $(2, 1) \in / R$ . Also $R$ is transitive as $x \leq y, y \leq z \Rightarrow x \leq z$ .

Q. Let x,y∈I and suppose that a relation R on I is defined by xRy if and only if x≤y then

R is reflexive but not symmetric

R is an equivalence relation

R is neither reflexive nor symmetric

R is symmetric but not transitive