Key Concepts and Formulas

• Set Difference: For two sets $A$ and $B$, $A - B = \{x \mid x \in A \text{ and } x \notin B\}$. In the context of relations, $R_1 - R_2$ contains ordered pairs that are in $R_1$ but not in $R_2$.
• Relation on a Set: A relation $R$ on a set $S$ is a subset of $S \times S$. This means for any pair $(a, b) \in R$, both $a$ and $b$ must belong to $S$.
• Prime Numbers: A natural number greater than 1 that has no positive divisors other than 1 and itself.

Step-by-Step Solution

Step 1: Understand the Definitions of $R_1$ and $R_2$ and the Universal Set

The universal set is $A = \{1, 2, \ldots, 50\}$. The relations are defined on $A$. This means for any pair $(x, y)$ to be in $R_1$ or $R_2$, both $x$ and $y$ must be elements of $A$.

$R_1 = \{(p, p^n) \mid p \text{ is a prime and } n \ge 0 \text{ is an integer}\}$ For $(p, p^n) \in R_1$, we must have:

1. $p$ is a prime number.
2. $p \in \{1, 2, \ldots, 50\}$.
3. $n \ge 0$ is an integer.
4. $p^n \in \{1, 2, \ldots, 50\}$, which implies $p^n \le 50$.

$R_2 = \{(p, p^n) \mid p \text{ is a prime and } n = 0 \text{ or } 1\}$ For $(p, p^n) \in R_2$, we must have:

1. $p$ is a prime number.
2. $p \in \{1, 2, \ldots, 50\}$.
3. $n=0$ or $n=1$.
4. $p^n \in \{1, 2, \ldots, 50\}$, which implies $p^n \le 50$.

Step 2: Determine the Conditions for Elements in $R_1 - R_2$

We are looking for elements $(p, p^n)$ such that $(p, p^n) \in R_1$ and $(p, p^n) \notin R_2$. This means the pair must satisfy the conditions for $R_1$ but not the conditions for $R_2$.

The conditions for $(p, p^n) \in R_1 - R_2$ are:

1. $p$ is a prime number.
2. $p \in \{1, 2, \ldots, 50\}$.
3. $n \ge 0$ is an integer (from $R_1$).
4. $p^n \in \{1, 2, \ldots, 50\}$ (from $R_1$).
5. It is NOT the case that ($n=0$ or $n=1$) (from $R_2$). This implies $n$ must be an integer such that $n \ge 2$.

So, we need to find pairs $(p, p^n)$ where $p$ is a prime number, $p \le 50$, $n$ is an integer with $n \ge 2$, and $p^n \le 50$.

Step 3: List all Prime Numbers $p \le 50$

The prime numbers less than or equal to 50 are: $P = \{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47\}$. There are 15 such primes.

Step 4: Enumerate the Pairs $(p, p^n)$ satisfying the conditions for $R_1 - R_2$

We check each prime $p$ for possible values of $n \ge 2$ such that $p^n \le 50$.

• For $p=2$: We need $2^n \le 50$ with $n \ge 2$.

  • $n=2: 2^2 = 4 \le 50$. Pair: $(2, 4)$.
  • $n=3: 2^3 = 8 \le 50$. Pair: $(2, 8)$.
  • $n=4: 2^4 = 16 \le 50$. Pair: $(2, 16)$.
  • $n=5: 2^5 = 32 \le 50$. Pair: $(2, 32)$.
  • $n=6: 2^6 = 64 > 50$. No more values for $n$. There are 4 pairs for $p=2$.

• For $p=3$: We need $3^n \le 50$ with $n \ge 2$.

  • $n=2: 3^2 = 9 \le 50$. Pair: $(3, 9)$.
  • $n=3: 3^3 = 27 \le 50$. Pair: $(3, 27)$.
  • $n=4: 3^4 = 81 > 50$. No more values for $n$. There are 2 pairs for $p=3$.

• For $p=5$: We need $5^n \le 50$ with $n \ge 2$.

  • $n=2: 5^2 = 25 \le 50$. Pair: $(5, 25)$.
  • $n=3: 5^3 = 125 > 50$. No more values for $n$. There is 1 pair for $p=5$.

• For $p=7$: We need $7^n \le 50$ with $n \ge 2$.

  • $n=2: 7^2 = 49 \le 50$. Pair: $(7, 49)$.
  • $n=3: 7^3 = 343 > 50$. No more values for $n$. There is 1 pair for $p=7$.

• For $p \ge 11$: For any prime $p \ge 11$, the smallest possible value for $n \ge 2$ is $n=2$.

  • If $p=11$, $p^2 = 11^2 = 121 > 50$. Since $11^2 > 50$, for any prime $p > 11$ and any $n \ge 2$, $p^n$ will also be greater than 50. Therefore, there are no pairs for primes $p \ge 11$.

Step 5: Count the Total Number of Elements in $R_1 - R_2$

The total number of elements in $R_1 - R_2$ is the sum of the counts for each prime: Number of elements = (Count for $p=2$) + (Count for $p=3$) + (Count for $p=5$) + (Count for $p=7$) Number of elements = $4 + 2 + 1 + 1 = 8$.

Common Mistakes & Tips

• Forgetting the $p^n \le 50$ constraint: This is crucial for defining the actual elements within the given set.
• Misinterpreting $n \ge 0$ vs. $n \ge 2$: The difference between $R_1$ and $R_2$ lies precisely in the allowed values of $n$. Elements in $R_1 - R_2$ must have $n \ge 2$.
• Not checking all primes: While larger primes quickly exceed the bound, it's important to systematically check primes until their squares exceed 50.
• Including pairs where $p^n=1$: The condition $p^n \le 50$ includes $p^0=1$. However, these pairs $(p, 1)$ are in $R_2$ (for $n=0$), so they are not in $R_1 - R_2$.

Summary

To find the number of elements in $R_1 - R_2$, we identified that these are ordered pairs $(p, p^n)$ where $p$ is a prime number, $p \le 50$, $n$ is an integer with $n \ge 2$, and $p^n \le 50$. By systematically checking each prime number less than or equal to 50, we enumerated all such pairs: $(2,4), (2,8), (2,16), (2,32)$ for $p=2$; $(3,9), (3,27)$ for $p=3$; $(5,25)$ for $p=5$; and $(7,49)$ for $p=7$. Summing these counts gives a total of 8 elements.

The final answer is $\boxed{8}$.