Chapter 3.5, Problem 50PPE

To determine

To calculate: the numbers of subsets that the set has, without listing each subset of a set.

Answer to Problem 50PPE

The numbers of subsets that the set has: $\left[2^6=64\right]$

Explanation of Solution

Given information:

Without listing each subset of a set, determine the number of subsets that the set has.

Calculation:

Consider a set $A=\left\{1\right\}$ that contains one element.

Then the subsets of $A$ are $\varnothing$ and $\left\{1\right\}$

So, the number of subsets is $2=2^1$

Consider a set $B=\{1,2\}$ that contains two elements.

Then the subsets of $B$ are $\varnothing ,\left\{1\right\},\left\{2\right\}$ and $\left\{1,2\right\}$

Here, the number of subsets is $4=2^2$

Consider a set that contains three elements.

Then the subsets of $A$ are $\varnothing ,\left\{1\right\},\left\{2\right\},\left\{3\right\},\{1,2\},\{1,3\},\{2,3\}$ and $\{1,2,3\}$

Here, the number of subsets is $8=2^3$ .

Similarly on continuing in this process, conclude that the number of subsets that a set can have is given by $2^n$ , where $n$ denotes the number of elements contained in that set.

For example, consider the set $A=\{1,2,3,4,5,6\}$

By the above statement, the numbers of subsets are: $\left[2^6=64\right]$ .