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 \(P \cup Q \)
Solution:
\(P=\{x:x\ \text{is a prime number and } 0< x\leq 20\}\)
\(P=\{2,3,5,7,11,13,17,19\}\)
\( Q=\{x:x\text{ is a divisor of } 210\ \text{and} \ 0< x\leq 20\}\)
\( Q=\{1,2,3,5,6,7,10,14,15\}\)
\(P \cup Q \)=\(\{2,3,5,7,11,13,17,19\}\)\(\cup\)\(\{1,2,3,5,6,7,10,14,15\}\)
\(P \cup Q \)=\(\{1,2,3,5,6,7,10,11,13,14,15,17,19\}\)
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.
Union \(\cup\) means combine all elements from both sets.
No repetition — each element appears only once, even if it’s in both sets.
Tips for Students:
To find the union, don’t repeat numbers.
Use color-coding or ticks to mark used numbers.