Question

If A = {1,2,3........,10} then number of subsets of A containing only odd number is

• A. 31
• B. 32
• C. 27
• D. 30

Answer 1

The Correct Option is A

We are given a set \( A = \{1, 3, 5, 7, 9\} \) containing 5 odd numbers. For each element in the set, we have two choices: either include it in the subset or exclude it. Since there are 5 elements, the total number of subsets is: \[ 2^5 = 32 \] However, we need to exclude the empty set, which does not contain any odd numbers. Therefore, the number of subsets of \( A \) that contain only odd numbers is: \[ 32 - 1 = 31 \] Thus, the correct answer is (A) 31.

Answer 2

Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

We are looking for the number of subsets of A that contain only odd numbers.

First, identify the odd numbers in set A. Let O be the set of odd numbers in A:

\[ O = \{1, 3, 5, 7, 9\} \]

The number of odd elements in A is the number of elements in set O, which is \(n(O) = 5\).

A subset of A containing only odd numbers must be a subset of the set O.

The number of subsets of a set with \(k\) elements is given by the formula \(2^k\).

In this case, the set O has \(k=5\) elements.

Therefore, the number of subsets of O (which are the subsets of A containing only odd numbers) is:

\[ \text{Number of subsets} = 2^{n(O)} = 2^5 \]

\[ 2^5 = 2 \times 2 \times 2 \times 2 \times 2 -1= \mathbf{31} \]

Thus, there are 31 subsets of A containing only odd numbers.

Comparing this with the given options, the correct option is:

32