Question

2. If A is a finite set containing ‘n’ elements then find the number of subsets of A. ​

Answer 1

$\underline{ \cal{ \pmb{Given:}}}$

• A is finite set.
• It has ‘n’ number of elements.

$\underline{ \cal{ \pmb{To\:find:}}}$

• The number of subsets of A.

$\underline{ \cal{ \pmb{Required\:answer:}}}$

‎ ‎ ‎ ‎ ‎ ‎ The number of subsets of the finite set A equals $\underline{\boxed{\bf{\red{2^n}}}}$. For example, consider that B is a finite set with elements 1, 2 and 3. The number of subsets of B is 8 as:

$\: \: \: \: \: \: \rightarrow{ \sf{ {2}^{n} = {2}^{3} = 2 \times 2 \times 2 = \underline{\bf{8} }}}$

Let's verify our answer by writing all the subsets in the order! Remember, empty set is a subset of every set and every set is a subset of itself.

$\sf{A = \{1,2,3\}} \\ \downarrow \\ \sf{ A= \{ \emptyset\} \: \{ 1 \} \: \{2 \} \: \{3 \} \: \{1,2 \} \: \{1,3 \} \: \{2,3 \} \: \{1,2,3 \}}$

Now, count all the subsets. We get 8, hence our answer is correct!

______________________________________

$\underline{ \cal{ \pmb{Additional\: information:}}}$

Sets:

‎ ‎ ‎ ‎ ‎ ‎A set is a well-defined collection of objects. Let's understand what we mean by a well-defined collection of objects! We say that a collection is well-defined if there is some reason or rule by which we can say whether a given object of the universe belongs to or doesn't belong to the collection. To understand this better, let's look at some examples!

1. Let's consider the collection of natural numbers less than or equal to 10. In this example, we can definitely say what the collection is. The collection consists of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
2. Now, consider the collection of intelligent girls in a class. In this example, we can't say precisely which girls of the class belong to our collection. So, this collection is not well-defined.

Hence, the first collection is a set, whereas the second collection is not a set. In the first example, given above, the set of the first 10 natural numbers can be represented as A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

Elements of set:

‎ ‎ ‎ ‎ ‎ ‎The objects in a set are called it's elements or members. We usually denote the sets by capital letters A, B, C or X, Y, Z, etc.

• If a is an element of a set A, then we say that a belongs to A and we write it as a ∈ A.
• If a is not an element of a set A, then we say that a doesn't belongs to A and we write it as a ∉ A.

Representation of sets:

‎ ‎ ‎ ‎ ‎ ‎We can represent sets by the following methods:

1. Roaster method: In this method, a set is described by listing out all the elements in the set. For example, Let F be the set of all vowels in English alphabet. Then, we represent F as F = {a, e, i, o, u}.
2. Set-builder method: In this method, a set is described by using some property common to all its elements. The above mentioned example is represented as F = {x:x is a vowel in English alphabet}, in this method.

_____________________________________