Subsets

A is a subset of B if and only if every element of A is in B. Symbolically we write $A \subset B$ (means A is a subset of B).

For example, if $A = \{1, 2, 3, 4\}$, $B = \{1, 2, 3\}$ and $C = \{1, 2, 3, 4\}$ then, $B$ is a proper subset of $A$ i.e. $B \subset A$ and $C$ is an improper subset of $A$ i.e. $C \subseteq A$.

Important note: an empty set is a subset of any set.

If every element in $A$ is also in $B$, and there exist at least one element in $B$ that is not in $A$, we say that $A$ is a proper subset of $B$.

And if every element in $A$ is in $B$, and there is no element in $B$ that is not in $A$, we say that $A$ is an improper subset of $B$ and we write $A = B$ or symbolically we write $A \subseteq B$ or $B \subseteq A$.

Consider an example below:

When you look at the table, you will see that the number of subsets can be obtained by $2$ raised to the number of elements of the set under consideration. Therefore, the formula for finding the number of subsets of a set with $n$ elements is given by $2^n$, where $n$ is the number of elements of a set.

How many subsets are there in set $A = \{\text{Red}, \text{White}, \text{Yellow}\}$? List them.

Solution:

Case 1: Number of subsets

Set $A$ has $3$ elements.

Number of subsets $= 2^n$, so number of subsets of set $A = 2^3 = 8$.

Therefore set $A$ has $8$ subsets.

Case 2: List of subsets.

The subsets of set $A$ are: $\{\}$, $\{\text{Red}\}$, $\{\text{White}\}$, $\{\text{Yellow}\}$, $\{\text{Red},\text{White}\}$, $\{\text{Red},\text{Yellow}\}$, $\{\text{White},\text{Yellow}\}$, $\{\text{Red},\text{White},\text{Yellow}\}$.