$\text{A}$ is a subset of $\text{A}$ ($\text{A}\subseteq \text{B}$) if and only if every element of $\text{A}$ is in $\text{B}$

$\text{A}$ is a proper subset of $\text{A}$ ($\text{A}\subset \text{B}$) if and only if every element in $\text{A}$ is also in $\text{B}$,

and there exists at least one element in $\text{B}$ that is not in $\text{A}$

Example 1:

$\text{A}=\{1,2,3\}$

$\text{B}=\{3,1,2\}$

$\text{A}$ is a subset of $\text{A}$. ($\text{A}\subseteq \text{B}$)

Example 2:

$\text{P}=\{5,10,15,20\}$

$\text{Q}=\{5,10,15,20,25,30\}$

$\text{P}$ is a proper subset of $\text{Q}$ because the element 25 and 30 is not in the set $\text{P}$. ($\text{P}\subset \text{Q}$)

The empty set is therefore a proper subset of any non empty set

Total Number of Subsets

The total number of subsets is the number of sets with 0 elements, 1 elements, 2 elements, etc

The total number of subsets of a set with n elements is $2^n$

Example 1:

From the set below, list all the subsets and find the total number of subsets

$\text{Q}=\{\text{a, b, c, d}\}$

Subsets of $\text{Q}$ are:

$\{\},\{\text{a}\},\{\text{b}\},\{\text{c}\},\{\text{d}\}\{\text{a,b}\},\{\text{a, c}\}$

$\{\text{a, d}\},\{\text{b, c}\},\{\text{b, d}\},\{\text{c, d}\},\{\text{a, b, c}\},\{\text{a, b, d}\}\{\text{b, c ,d}\},\{\text{a, c, d}\},\{\text{a, b, c, d}\}$

Total number of subsets is $2^n=2^4=16$

Example 2:

Find the number of subsets of  $\text{P}=\{\text{a, b, c, d, e, f}\}$  that contain exactly 3 elements

The total number of subsets of $\text{P}$ is $2^6$

Using Pascal's triangle:

So, the number of subsets of $\text{P}$ that contain exactly 3 elements is 20