Subsets and Power Sets

We need a set to define its subset or power set. For example, once we have a set $T$, we can define what we mean by a subset of $T$and, similarly, what we mean by the power set of $T$.

Subsets

We call $S$ a subset of the set $T$ if all the members of $S$ are also members of $T$. We normally denote this as $S\sube T$. If $S$ is not a subset of $T$, we write it as $S\not\subseteq T$.

Proper and improper subsets

There are two kinds of subsets: proper and improper. The set $S$ is said to be a proper subset of $T$ if there is at least one member of $T$ that is not a member of $S$. Otherwise, $S$ is an improper subset of $T$.

If $S$ is a proper subset of $T$, we write it as $S\sub T$or$S\subsetneq T$. But if $S$ is an improper subset of $T$, we simply write it as s evident from this definition:

• An empty set is a subset of every set.

• Every set is an improper subset of itself.

• The cardinality of any subset of set $T$ is less than or equal to the cardinality of $T$.

Let’s consider an example where we have a company’s office building with employees working in multiple departments. The set of all employees of the company is taken as the set $O$. However, the employees working in the departments of the company (i.e., IT, finance, and HR) make the subsets of the set $O$, and we can represent them with $S_{IT}$, $S_F$, and $S_{HR}$.