Let R₁ and R₂ be relations on the set {1, 2, ......., 50} such that
R₁ = (p, pⁿ) : p is a prime and n ≥ 0 is an integer and
R₂ = {(p, pⁿ) : p is a prime and n = 0 or 1}.
Then, the number of elements in R₁ - R₂ is _______________.

Solution

Maths sir · 3 steps

R2 only removes the n = 0 and n = 1 pairs, so just count prime powers with n ≥ 2 that stay within 50.

1. What is left. R₂ is exactly the part of R₁ with n = 0 or n = 1. So R₁ - R₂ holds the pairs (p, pⁿ) with n ≥ 2, and pⁿ must still be in {1, 2, …, 50}.
2. Which primes matter. We need p² ≤ 50, so p ≤ 7. Primes from 11 onwards give p² ≥ 121, which is too large.

   Prime p	Powers pⁿ ≤ 50 with n ≥ 2	Pairs
   2	4, 8, 16, 32 (next is 64)	4
   3	9, 27 (next is 81)	2
   5	25 (next is 125)	1
   7	49 (next is 343)	1

3. So. 4 + 2 + 1 + 1 = 8, so R₁ - R₂ has 8 elements.

The common slip: 7. Stopping the search at p = 5 misses the pair (7, 49) and gives 7.

The idea to keep. When one set is built from part of another, the difference is the leftover condition. Here that means n ≥ 2. Then list the powers prime by prime until you cross the upper limit.

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