Solution

The correct answer is A is not correct but R is correct.

Key Points

• Given relation: R ⊂ A × A, where A = ℕ × ℕ and ((a,b),(c,d)) ∈ R ⇔ a ≤ c or b ≤ d
• To be a partial order, relation must be Reflexive, Anti-symmetric, and Transitive.
• Reflexive: For any (a,b) ∈ A, we check (a,b) R (a,b) ⇒ a ≤ a or b ≤ b ⇒ True. Hence reflexive property holds.
• Anti-symmetric test: Take (1,2) and (2,1). (1,2) R (2,1) ⇒ 1 ≤ 2 (True) (2,1) R (1,2) ⇒ 1 ≤ 2 (True) But (1,2) ≠ (2,1). Hence anti-symmetry fails.
• Since anti-symmetry fails, relation is not a partial order. Therefore Assertion A is false.
• Reason R: Definition of partial order is correct. Hence Reason R is true.

Additional Information

• Partial Order Conditions:
  • Reflexive: aRa
  • Anti-symmetric: aRb and bRa ⇒ a=b
  • Transitive: aRb and bRc ⇒ aRc
• Conclusion:
  • Relation fails anti-symmetry ⇒ Not a partial order.
  • Hence option 4 is correct.