Question

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 ______.

Answer 1

Correct Answer: 8

To solve the problem, we need to determine the number of elements in the set difference R₁−R₂, where both relations are defined over primes not exceeding 50. Let's break it down step-by-step.
1. **Identify primes ≤ 50**: The set is {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47}. There are 15 primes in total.
2. **Define R₁**: R₁ ={(p, pⁿ) : p is a prime, n≥0}. For each prime p, (p, p⁰=1), (p, p¹=p), (p, p²), ... , (p, p^m) are included, where m is the largest integer for which p^m ≤ 50.
3. **Find maximum n**: Calculate this m for each prime.
Example:
 - For prime 2: 2⁵=32 (n=0,1,2,3,4,5).
 - For prime 3: 3³ =27 (n=0,1,2,3).
Repeat for others:
 - 5²=25, 7²=49, 11¹=11, 13¹=13, ... , 47¹=47.
4. **Count R₁ elements**: Compute number for each prime, sum them:
- (2:6), (3:4), (5:3), (7:3), (11:2), (13:2), (17:2), ..., (47:2). Total = 6+4+3+3+10×2 =40.
5. **Define R₂**: R₂ ={(p, pⁿ) : p is a prime, n=0 or 1}. For each prime, {1, p}.
6. **Count R₂ elements**: Total = 15×2 = 30.
7. **Set difference R₁−R₂**: Contains elements in R₁ but not in R₂.
8. **Compute |R₁−R₂|**: 40 − 30 = 10.
9. **Verify within range (8,8)**: Since 10 is outside the range, there may be an error or misinterpretation of expected results.
Thus, if following steps are correct, the number of elements in R₁−R₂ is calculated as 10, though stated range is (8,8), indicating discrepancy.