Correct : Set Theory

Reflexivity (Proposition P):
• A relation R on a set A is reflexive if, for every element x ∈ A, x R x holds.
• In this case, the elements are ordered pairs (a, b) from N x N.
• For R to be reflexive, for any (a, b) ∈ N x N, it must be true that (a, b) R (a, b).
• According to the definition of R, (a, b) R (a, b) if a ≤ a or b ≤ b.
• We know that for any natural number 'a', the condition 'a ≤ a' is always true.
• Similarly, for any natural number 'b', the condition 'b ≤ b' is always true.
• Since 'a ≤ a' is true, the compound condition 'a ≤ a or b ≤ b' is true (because an OR statement is true if at least one of its components is true).
• Therefore, for every (a, b) ∈ N x N, (a, b) R (a, b) holds.
• Thus, R is reflexive.
• Proposition P is true.
Transitivity (Proposition Q):
• A relation R on a set A is transitive if, for all x, y, z ∈ A, whenever x R y and y R z, then x R z must also hold.
• Here, x = (a, b), y = (c, d), and z = (e, f).
• We need to check if: [(a, b) R (c, d) AND (c, d) R (e, f)] → (a, b) R (e, f).
• Let's test with a counterexample. We aim to find (a, b), (c, d), and (e, f) such that (a, b) R (c, d) is true, (c, d) R (e, f) is true, but (a, b) R (e, f) is false.
• For (a, b) R (e, f) to be false, by definition, it means NOT (a ≤ e or b ≤ f), which is equivalent to (a > e AND b > f).
• Let's choose the following pairs from N x N (assuming N includes positive integers):
    Let (a, b) = (5, 5)
    Let (c, d) = (1, 10)
    Let (e, f) = (4, 4)
• Check (a, b) R (c, d):
    (5, 5) R (1, 10) means 5 ≤ 1 (False) or 5 ≤ 10 (True).
    Since (False or True) is True, (5, 5) R (1, 10) holds.
• Check (c, d) R (e, f):
    (1, 10) R (4, 4) means 1 ≤ 4 (True) or 10 ≤ 4 (False).
    Since (True or False) is True, (1, 10) R (4, 4) holds.
• Check if (a, b) R (e, f) holds:
    (5, 5) R (4, 4) means 5 ≤ 4 (False) or 5 ≤ 4 (False).
    Since (False or False) is False, (5, 5) R (4, 4) does NOT hold.
• We have found a scenario where (a, b) R (c, d) and (c, d) R (e, f) are both true, but (a, b) R (e, f) is false. This violates the definition of transitivity.
• Therefore, R is not transitive.
• Proposition Q is false.
Conclusion:
• Proposition P (R is reflexive) is true.
• Proposition Q (R is transitive) is false.
This corresponds to option B.