Question

Let relation defined as (x₁, y₁) R (x₂, y₂)

x₁ ≤ x₂, y₁ ≤ y₂ and given that

(a) R is reflexive but not symmetric. 

(b) R is transitive. then 

(1) (a) is true, (b) is false 

(2) (a) is false, (b) is true 

(3) Both are true 

(4) Both are false

Answer 1

Correct option is (3) Both are true 

(x₁, y₁) R (x₂, y₂) 

When x₁ ≤ x₂, y₁ ≤ y₂ 

For reflexive 

(x₁, y₁) R (x₁, y₁) 

⇒ x₁ ≤ x₁ & y₁ ≤ y₁ 

so, R is reflexive 

For symmetric,

When (x₁, y₁) R (x₂, y₂) 

⇒ x₁ ≤ x₂ & y₁ ≤ y₂ 

for (x₂, y₂) R (x₁, y₁) 

x₂ ≤ x₁ & y₂ ≤ y₁

That is not necessarily true so R is not symmetric 

For transitive,

If (x₁, y₁) R (x₂, y₂) ⇒ x₁ ≤ x₂ & y₁ ≤ y₂ 

& (x₂, y₂) R (x₃, y₃) ⇒ x₂ ≤ x₃ & y₂ ≤ y₃ 

For (x₁, y₁) R (x₃, y₃) ⇒ x₁ ≤ x₃ & y₁ ≤ y₃ 

So, R is transitive 

Both (a) & (b) are true.