if P = {x:x is odd, $-1 < x \leq 20$} and Q is {y:y is prime, $-2 < y \leq 25$, find P $\cap$ Q

1. Analyze the Question

We are given two sets, $P$ and $Q$, defined by specific conditions. We need to find the intersection of these two sets ($P \cap Q$), which consists of all elements that are present in both sets.

2. Explain the Concept

• Set P: Contains odd numbers $x$ such that $-1 < x \leq 20$.
• Set Q: Contains prime numbers $y$ such that $-2 < y \leq 25$.
• Intersection ($P \cap Q$): This means we are looking for numbers that are both odd and prime within the specified range ($-1 < x \leq 20$).

3. Show the Solution

Step 1: List the elements of Set P Set $P$ contains odd numbers greater than $-1$ and less than or equal to $20$. $P = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}$

Step 2: List the elements of Set Q Set $Q$ contains prime numbers greater than $-2$ and less than or equal to $25$. $Q = \{2, 3, 5, 7, 11, 13, 17, 19, 23\}$

Step 3: Find the intersection ($P \cap Q$) We look for numbers that appear in both lists. Note that $2$ is prime but not odd, so it is excluded.

• Is 1 in Q? No (1 is not a prime number).
• Is 3 in both? Yes.
• Is 5 in both? Yes.
• Is 7 in both? Yes.
• Is 9 in both? No (9 is not prime).
• Is 11 in both? Yes.
• Is 13 in both? Yes.
• Is 15 in both? No (15 is not prime).
• Is 17 in both? Yes.
• Is 19 in both? Yes.

Combining these, we get: $\{3, 5, 7, 11, 13, 17, 19\}$

4. Identify the Answer

Comparing our result with the given options, we see that our set matches option C.

5. Exam Tip

Always remember that 1 is not a prime number. Many students mistakenly include 1 in sets of prime numbers, which leads to errors in questions involving set theory or probability. Also, remember that 2 is the only even prime number; all other prime numbers are odd.