Question

A set contains nine elements.

a) How many subsets does it have?
b) How many proper subsets does it have?

Answer 1

Step-1:Subsets:

Set $A$ is said to be a subset of Set $B$ if all the elements of Set $A$ are also present in Set $B$. i.e. set $A$ is contained inside Set $B$.

Types of Subsets

• Proper Subset
• Improper Subsets

Step-2: Finding the number of subsets:

We know that, The number of elements in Set $A$ is $n$.

Then the number of subsets formed by Set $A$ is $2^n$.

Given, The number of elements in Set $A$ is $9$.

Therefore, The number of subsets formed by Set $A$ is $2^9=512$.

Step-3: Finding the number of proper subsets:

Set $A$ is considered to be a proper subset of Set $B$ if Set $B$ contains at least one element that is not present in Set $A$.

We know that,

The number of elements in Set $A$ is $n$.

Then the number of proper subsets formed by Set $A$ is $2^n-1$.

Given, The number of elements in Set $A$ is $9$.

Therefore, The number of proper subsets formed by Set $A$ is $2^9-1=512-1=511$.

Hence, the number of subsets is $512$ and the number of subsets proper subsets is $511$.