1. Understand the set and the relations

We are working on the set $A=\{1,2,3,\dots,50\}.$

The relations are: $R_1=\{(p,p^n): p\text{ is prime and } n\ge 0\text{ is an integer}\}$ and $R_2=\{(p,p^n): p\text{ is prime and } n=0\text{ or }1\}.$

Since these are relations on $A$, both coordinates of every ordered pair must lie in $A$.

So for a prime $p\le 50$, the pair $(p,p^n)$ belongs to $R_1$ only when $p^n\le 50.$

Also:

• $n=0 \Rightarrow p^0=1$, so $(p,1)\in R_2$.
• $n=1 \Rightarrow p^1=p$, so $(p,p)\in R_2$.

Thus, $R_1-R_2$ consists of those pairs in $R_1$ with $n\ge 2.$ So we need to count all ordered pairs $(p,p^n)$ such that:

• $p$ is prime,
• $n\ge 2$,
• $p^n\le 50$.

2. Check each prime $p\le 50$

The primes up to $50$ are: $2,3,5,7,11,13,17,19,23,29,31,37,41,43,47.$

We count powers $p^n\le 50$ for $n\ge 2$.

For $p=2$

$2^2=4,\quad 2^3=8,\quad 2^4=16,\quad 2^5=32,\quad 2^6=64>50.$ Valid values: $n=2,3,4,5$.

Pairs: $(2,4),(2,8),(2,16),(2,32)$ Count $=4$.

For $p=3$

$3^2=9,\quad 3^3=27,\quad 3^4=81>50.$ Valid values: $n=2,3$.

Pairs: $(3,9),(3,27)$ Count $=2$.

For $p=5$

$5^2=25,\quad 5^3=125>50.$ Valid value: $n=2$.

Pair: $(5,25)$ Count $=1$.

For $p=7$

$7^2=49,\quad 7^3=343>50.$ Valid value: $n=2$.

Pair: $(7,49)$ Count $=1$.

For $p\ge 11$

$11^2=121>50.$ So for any larger prime, $p^2>50$, hence no valid $n\ge 2$.

Count $=0$ for all remaining primes.

3. Total count

Adding all valid pairs: $4+2+1+1=8.$

Therefore, $|R_1-R_2|=8.$

4. Compare with stored answer

Stored correct answer: $8$

Our derived answer: $8$

They match.