Question

Let \[S{\text{ }} = {\text{ }}\left\{ {1,2,3....10} \right\}.\]The number of subsets of S containing only odd number is
A. 15
B. 31
C. 63
D. 7

Answer 1

Hint: The given question is solved by using set theory. Instead there is a subset of it. Let us discuss the set and their subset on it. In a set, there are elements or numbers.
Set: Set is a well-defined collection of distinct objects or elements. The arrangement of the objects/elements in the set does not matter.

Complete step by step answer:
Even number: The number which is divisible by $2$ is called an even number like \[A \cup B\] \[2,4,6,8....\] so on.
Odd number: The number which is not divisible by $2$ is called an odd number like \[3,9,11...\] so on.
Sets are the well-defined collection of distinct elements. Sets contain the intersection and the union of the elements.
The intersection of two sets: Let A and B be two sets and their intersection will be like \[A \cap B\]. The symbol of intersection \[ \cap \].
Union of two sets: Let A and B be two sets and their union be like \[A \cup B\]. The symbol of a union is like \[ \cup \].
Subset: is that set in which all the elements are contained in another set. In the given question set is \[S{\text{ }} = {\text{ }}\left\{ {1,2,3...10} \right\}\]
\[S\] is set containing numbers \[1\] to \[10.\]
In which the odd numbers are \[1,3,5,7,9...\]
So the number subset is five containing only numbers.
The identity of sets is \[{2^n} - 1...\]

Where \[n\] is equal to \[5\]
We have \[5\] odd numbers...
So the identity becomes
= \[{2^5} - 1 - - - - - - (A)\]
Like \[{2^5}\] be \[2 \times 2 \times 2 \times 2 \times 2\]which is equal to \[32\].
So equation \[A\] becomes
= \[32 - 1\]
= \[31\]

There are \[31\] subsets in it.

Note: The given question is based on set theory. For solving the question using \[2 - 1\]identity on it. The elements in a set are odd in the given question. So in that way, we can find the subsets of the set containing only odd numbers. The elements in a set are randomly arranged or we can say that no arrangement does not matter.