Question

Let \( S = \{1, 2, 3, \dots, 10\} \). The number of subsets of \( S \) containing only odd numbers is:

• A. \( 15 \)
• B. \( 31 \)
• C. \( 63 \)
• D. \( 7 \)
• E. \( 5 \)

Hint:

In set theory, "only odd numbers" means elements are restricted to the odd set. If the empty set \( \emptyset \) is included, the answer is 32; if not, it is 31. Usually, multiple-choice options guide this choice.

Answer 1

The Correct Option is B

Concept: A subset containing "only" odd numbers must be constructed using only the elements from the set of odd numbers available in \( S \). The number of all possible subsets of a set with \( m \) elements is \( 2^m \). If we exclude the empty set (which contains no numbers), we subtract 1.

Step 1: Identify the odd numbers in \( S \).
Odd numbers in \( \{1, 2, 3, \dots, 10\} \) are \( \{1, 3, 5, 7, 9\} \). Number of odd elements \( m = 5 \).

Step 2: Calculate the number of non-empty subsets.
Total subsets of the odd set \( = 2^5 = 32 \). These subsets consist of only odd numbers. If the question implies non-empty subsets (standard in this context): \[ 2^5 - 1 = 31 \]