Question

On ℝ , consider the relation R defined as xRy if and only

if x ≤ y. Determine whether R is Reflexive , symmetric,

Antisymmetric and Transitive?​

Answer 1

Answer:

it is reflexive and transitive

take some examples like 11 22 33 12 21 23 32 13 31

Answer 2

SOLUTION

TO DETERMINE

On ℝ , consider the relation R defined as xRy if and only if x ≤ y.

Determine whether R is Reflexive , symmetric, Antisymmetric and Transitive?

EVALUATION

On ℝ , consider the relation R defined as xRy if and only if x ≤ y

CHECKING FOR REFLEXIVE

Let x ∈ ℝ

Then x ≤ x

So we have (x, x) ∈ R

So R is Reflexive

CHECKING FOR SYMMETRIC

Since 2 < 3 then (2,3) ∈ R

But (3,2) ∉ R

So R is not symmetric

CHECKING FOR TRANSITIVE

Let x , y , z ∈ ℝ

Also let (x,y) ∈ R and (y,z) ∈ R

⇒ x ≤ y and y ≤ z

⇒ x ≤ z

⇒(x,z) ∈ R

Thus (x, y) ∈ R and (y, z) ∈ R implies (x, z) ∈ R

R is transitive

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