Question 3: consider the sets \(P=\{x:x\ \text{is a prime number and } 0< x\leq 20\}\) and \( Q=\{x:x\text{ is a divisor of } 210\ \text{and} \ 0< x\leq 20\}\)
\((i)\) \(\;\) Find \(P \cap Q \)
Solution:
\(P=\{x:x\ \text{is a prime number and } 0< x\leq 20\}\)
\(P=\{2,3,5,7,11,13,17,19\}\)
\( Q=\{x:x\text{ is a divisor of } 210\ \text{and} \ 0< x\leq 20\}\)
\( Q=\{1,2,3,5,6,7,10,14,15\}\)
\(P \cap Q \)=\(\{2,3,5,7,11,13,17,19\}\)\(\cap\)\(\{1,2,3,5,6,7,10,14,15\}\)
\(P \cap Q \)=\(\{2,3,5,7\}\)
Explanation:
\(P=\{x:x\ \text{is a prime number and } 0< x\leq 20\}\)
Write all the prime numbers greater than \(0\) and less than or equal to \(20\).
\( Q=\{x:x\text{ is a divisor of } 210\ \text{and} \ 0< x\leq 20\}\)
To find the divisors of \(210\) up to \(20\), start dividing \(210\) by the numbers \(1, 2, 3, …, 20\).
Use a calculator or long division, and select only those numbers that divide exactly — meaning they leave no remainder.
Intersection \(\cap\) means the common elements in both sets.
Tips for Students:
- Use a highlighter or circle common elements in both lists.
- Always write sets in increasing order for easier comparison.