Given U = {x: x is a positive integer less than 15} and P = {x: x is even number from 1 to 14}. Find the compliment of P.

1. Analyze the question

We are given a Universal set ($U$) containing positive integers less than 15, and a subset ($P$) containing even numbers from 1 to 14. We need to find the complement of $P$ (denoted as $P'$ or $P^c$), which consists of all elements in $U$ that are not in $P$.

2. Explain the concept

• Universal Set ($U$): The set of all elements under consideration. Here, "positive integers less than 15" means $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14\}$.
• Subset ($P$): The set of even numbers from 1 to 14. These are $\{2, 4, 6, 8, 10, 12, 14\}$.
• Complement ($P'$): This is the set of elements that are in $U$ but are not in $P$. Essentially, you remove all even numbers from the universal set, which leaves you with the odd numbers.

3. Show the solution

• Step 1: List the elements of $U$ $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14\}$

• Step 2: List the elements of $P$ $P = \{2, 4, 6, 8, 10, 12, 14\}$

• Step 3: Find $P'$ (Elements in $U$ not in $P$) We remove $\{2, 4, 6, 8, 10, 12, 14\}$ from $U$: $P' = \{1, 3, 5, 7, 9, 11, 13\}$

4. Identify the answer

Comparing our result $\{1, 3, 5, 7, 9, 11, 13\}$ with the given options:

• Option A includes 15 (which is not in $U$ because $U$ is less than 15).
• Option B includes 2 (which is in $P$).
• Option C matches our result exactly.
• Option D is missing several odd numbers.

5. Tip

When dealing with set complements, always double-check the definition of your Universal set. If the question says "less than 15," 15 is excluded. If it says "15 or less," 15 is included. Always list out the elements of the Universal set first to avoid missing numbers!