Let R_(1) and R_(2) be relations on the set {1,2,dots,50} such that R_(1)={(p,p^(n)):} : is a prime and n = 0 is an integer } and R_(2)={(p,p^(n)):p:} is a prime and n=0 or 1 }. Then, the number of elements in R_(1)-R_(2) is qquad

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Let  R_(1) and  R_(2) be relations on the set  {1,2,dots,50} such that  R_(1)={(p,p^(n)):} : is a prime and  n >= 0 is an integer  } and  R_(2)={(p,p^(n)):p:} is a prime and  n=0 or 1 }. Then, the number of elements in  R_(1)-R_(2) is  qquad

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Define R_(1) and R_(2): R_(1) includes pairs (p, p^n) where p is prime and n is any non-negative integer. R_(2) includes pairs (p, p^n) where p is prime and n is 0 or 1.Find pairs in R_(1) - R_(2): To find R_(1) - R_(2), we need to count the pairs in R_(1) that are not in R_(2). This means we're looking for pairs where n is greater than 1.Identify primes in set: The primes in the set {1,2,...,50} are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47. That's 15 primes.Calculate powers of primes: For each prime p, there are infinite values of n where n > 1, but since we're limited to the set {1,2,...,50}, we need to count the powers of p that are less than or equal to 50.Example with prime 2: Let's take an example: for p = 2, the powers are 2^2, 2^3, ..., up to the largest power of 2 that is less than or equal to 50. That's 2^5 = 32. So we have 2^2, 2^3, 2^4, and 2^5.Repeat for each prime: We repeat this process for each prime, but we must be careful not to count any power of a prime that is greater than 50.Final count of powers: After calculating, we find that the powers of primes less than or equal to 50 for each prime are: 2^2, 2^3, 2^4, 2^5, 3^2, 3^3, 3^4, 5^2, 5^3, 7^2, 7^3, 11^2, 13^2, 17^2, 19^2, 23^2, 29^2, 31^2, 37^2, 41^2, 43^2, and 47^2.Counting these, we get 4 powers of 2, 3 powers of 3, 2 powers of 5, 2 powers of 7, and 1 power each for the rest of the primes. That's 4 + 3 + 2 + 2 + 11 = 22 elements.

Let $R_{1}$ and $R_{2}$ be relations on the set \{ $1$ , $2$ ,\dots, $50$ \} such that $R_{1}=\{(p,p^{n})\}: p$ is a prime and $n \geq 0$ is an integer } and $R_{2}=\{(p,p^{n})\}:p$ is a prime and $n=0$ or $1$ . Then, the number of elements in $R_{1}-R_{2}$ is ______.

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