Let R1 and R2 be relations on the set {1, 2, ..., 50} such that R1 = {(p, pⁿ) : p is a prime and n ≥

Q: Why do we only consider $n \geq 2$?

Great question! The set $R_1$ includes all $n \geq 0$. The set $R_2$ includes $n=0$ and $n=1$. When we perform the set difference $R_1 - R_2$, we are removing all elements present in $R_2$ from $R_1$. Thus, we are left with only those pairs where $n$ is strictly greater than $1$, meaning $n \geq 2$.

Q: Do I need to check if $p$ is a composite number?

The definition of the relation explicitly states that $p$ must be a prime number. Therefore, we only test $p \in \{2, 3, 5, 7, 11, \dots\}$. Composite numbers like $4$ or $6$ are not primes, so they are excluded from the relation entirely.

Q: What happens if $p^n$ is exactly 50?

The set $S$ is defined as $\{1, 2, \dots, 50\}$, which includes 50. If $p^n$ were equal to 50, it would be a valid element. However, since $p$ is prime, $p^n$ can only be 50 if $p=2$ and $n$ is not an integer, or if $p=5$ and $n$ is not an integer. Since $n$ must be an integer, $p^n$ will never be 50 for any prime $p$.

Question

Let $R_1$ and $R_2$ be relations on the set $\{1, 2, \dots, 50\}$ such that $R_1 = \{(p, p^n) : p 	ext{ is a prime and } n \geq 0 	ext{ is an integer}\}$ and $R_2 = \{(p, p^n) : p 	ext{ is a prime and } n = 0 	ext{ or } 1\}$. Then, the number of elements in $R_1 - R_2$ is ________.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are working within the set $S = \{1, 2, \dots, 50\}$. We define two relations, $R_1$ and $R_2$, based on prime powers $p^n$.

$R_1$ consists of all pairs $(p, p^n)$ where $n \geq 0$. $R_2$ acts as a filter that includes only those pairs where $n=0$ or $n=1$.

Our objective is to find the number of elements in the set $R_1 - R_2$. This implies we are looking for all pairs $(p, p^n)$ such that $n \geq 2$ and $p^n \leq 50$.

### The Systematic Search

We perform a search for all primes $p$ and exponents $n \geq 2$ that satisfy the condition $p^n \leq 50$.

For $p=2$:
We test the powers $2^n$:
$$2^2 = 4, \quad 2^3 = 8, \quad 2^4 = 16, \quad 2^5 = 32$$
Since $2^6 = 64 > 50$, we have **4 valid pairs** for $p=2$.

For $p=3$:
We test the powers $3^n$:
$$3^2 = 9, \quad 3^3 = 27$$
Since $3^4 = 81 > 50$, we have **2 valid pairs** for $p=3$.

For $p=5$:
We test the powers $5^n$:
$$5^2 = 25$$
Since $5^3 = 125 > 50$, we have **1 valid pair** for $p=5$.

For $p=7$:
We test the powers $7^n$:
$$7^2 = 49$$
Since $7^3 = 343 > 50$, we have **1 valid pair** for $p=7$.

### The Boundary of Possibility

We must consider if any primes $p \geq 11$ can yield a valid pair.

For $p=11$, the smallest power with $n \geq 2$ is:
$$11^2 = 121$$
Because $121 > 50$, any prime $p \geq 11$ will result in a value greater than 50 for $n \geq 2$. Thus, no further pairs exist.

### The Final Tally

To find the total number of elements in $R_1 - R_2$, we sum the valid pairs found for each prime:

$$Total = 4 + 2 + 1 + 1 = 8$$

We have successfully navigated the constraints and determined that there are exactly **8 elements** in $R_1 - R_2$.

Enter Numerical Value:

Visualized Solution

Defining the Set and Relations

Condition for $R_{1} - R_{2}$

Case 1: Prime $p = 2$

Upper Bound for $p = 2$

Case 2: Prime $p = 3$

Case 3: Primes $p = 5$ and $p = 7$

Checking Primes $p \geq 11$

Final Summation

The Sigma Insight: Domain and Range of a Relation

Analyzing the Setup

The Systematic Search

The Boundary of Possibility

The Final Tally

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Enter Numerical Value:

Visualized Solution

Defining the Set and Relations

Condition for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R1​−R2​

Case 1: Prime .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } p=2

Upper Bound for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } p=2

Case 2: Prime .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } p=3

Case 3: Primes .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } p=5 and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } p=7

Checking Primes .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } p≥11