Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) ∈ R if and only if max{x, y} ∈ {3, 4}. Then among the statements
(S₁): The number of elements in R is 18, and
(S₂): The relation R is symmetric but neither reflexive nor transitive

Solution

Maths sir · 3 steps

Count pairs by their maximum: max exactly 3, plus max exactly 4. Then test each property with one pair.

1. Max equal to 3. Both entries come from {0, 1, 2, 3}, but not both from {0, 1, 2}. That gives 4² - 3² = 16 - 9 = 7 pairs.
2. Max equal to 4. Both entries come from {0, 1, 2, 3, 4}, but not both from {0, 1, 2, 3}. That gives 5² - 4² = 25 - 16 = 9 pairs. So n(R) = 7 + 9 = 16, not 18, and S₁ is false.

   Property	Test	Result
   Reflexive	(0, 0): max{0, 0} = 0 ∉ {3, 4}	fails
   Symmetric	max{x, y} = max{y, x} for every pair	holds
   Transitive	(1, 3), (3, 1) ∈ R but (1, 1) ∉ R	fails

3. So. R is symmetric but neither reflexive nor transitive, so S₂ is true. S₁ is false, so the answer is (c).

The idea to keep. To count pairs with max{x, y} = k, subtract: (k+1)² - k² when the set starts at 0. To break transitivity, go out to a big element and come back to a small one.

Answer from the NTA final answer key. Solution by PrepWiser.