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For real numbers x and y, define xRy if and only if x - y + π is an irrational number. Then the relation R is

For real numbers x and y, define xRy if and only if x - y + π is an irrational number. Then the relation R is Correct Answer Reflexive

Concept:

A relation R in A is said to be:

  • Reflexive if aRa, ∀ a ∈ A.
  • Symmetric if aRb ⇒ bRa, ∀ a, b ∈ A.
  • Transitive if aRb and bRc ⇒ aRc, ∀ a, b, c ∈ A.
  • Any relation which is reflexive, symmetric and transitive is called an equivalence relation.

Calculation:

Let's check the given relation for its type one by one.

Reflexive: xRx = x - x + π = π, which is irrational. Therefore, R is reflexive.

Symmetric: Let's say that xRy is true i.e. x - y + π is irrational. Now, yRx = y - x + π will not always be irrational.

For x = 2π, y = π 

⇒ x - y + π =   2π - π + π =  2π → Irrational number 

⇒ y - x + π =   π - 2π + π =  0 →  rational number 

So, the relation is not symmetric.

Transitive: If π R 1  and 1 R 2π are true, then π R 2π = π - 2π + π = 0 is rational. Therefore, R is not transitive.

Since the relation R is reflexive but not symmetric and transitive.

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