Determine the number of proper subsets contained in Q if Q={x|||xis an even, positive integer andx<13}

We are given the set \( Q \), which consists of even, positive integers less than 13. To determine the number of proper subsets of this set, let's follow these steps:

### Step 1: Identify the elements of set \( Q \)
The set \( Q \) includes all even positive integers less than 13. These integers are:
\[
Q = \{2, 4, 6, 8, 10, 12\}
\]

So, \( Q \) has 6 elements.

### Step 2: Total number of subsets
The number of subsets of a set with \( n \) elements is given by the formula:
\[
\text{Total subsets} = 2^n
\]
where \( n \) is the number of elements in the set. Since \( Q \) has 6 elements, the total number of subsets is:
\[
2^6 = 64
\]

### Step 3: Number of proper subsets
A **proper subset** is any subset that is not equal to the set itself. Therefore, to find the number of proper subsets, we subtract 1 from the total number of subsets (since the set itself is not a proper subset):
\[
\text{Proper subsets} = 64 - 1 = 63
\]

### Final Answer:
The number of proper subsets of \( Q \) is **63**.

Would you like more details or have any questions?

Here are some related questions for practice:
1. How many subsets does a set with 8 elements have?
2. What is the number of proper subsets of a set with 5 elements?
3. If a set has 7 elements, how many improper subsets does it have?
4. How would the number of subsets change if \( Q \) had 10 elements?
5. Can a set with 0 elements have proper subsets?

**Tip:** Remember, the number of proper subsets of a set is always one less than the total number of subsets.

Determine the number of proper subsets contained in Q if Q={x|||x is an even, positive integer and x<13}.

Learn how to determine the number of proper subsets of the set Q, which contains even positive integers less than 13. This solution uses set theory concepts and the formula for calculating total and proper subsets. Suitable for students in Grades 6-8.

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