Sets - Subsets, Power Set, Universal Set

Definition of a Subset: A set $A$ is said to be a subset of a set $B$ if every element of $A$ is also an element of $B$. We write this as $A \subseteq B$. If $A$ is not a subset of $B$, we write $A \not\subseteq B$. Visually, this is represented in a Venn Diagram as a small circle $A$ residing entirely within the perimeter of a larger circle $B$.

Proper Subsets and Supersets: If $A \subseteq B$ and $A \neq B$, then $A$ is called a proper subset of $B$, denoted by $A \subset B$. In this relationship, $B$ is called the superset of $A$. Visually, this means that while $A$ is inside $B$, there is some shaded area in $B$ that does not belong to $A$, indicating $B$ has elements not present in $A$.

The Empty Set and Itself: Every set $A$ is a subset of itself ($A \subseteq A$). Additionally, the empty set $\emptyset$ (or null set) is considered a subset of every set. Visually, the empty set is like an empty container that can be placed inside any other container.

Power Set: The collection of all subsets of a set $A$ is called the power set of $A$, denoted by $P(A)$. Every element of the power set is a set itself. For example, if $A$ is represented by a box of distinct items, the power set is a master list of every possible combination of items you could take from that box, including taking nothing and taking everything.

Universal Set: In a particular context, all sets under consideration are treated as subsets of a larger fixed set called the Universal Set, denoted by $U$. In Venn Diagrams, the Universal Set is visually represented by a large outer rectangle that serves as the 'universe' for all other set circles drawn inside it.

Subsets of Real Numbers (Intervals): The set of Real Numbers $\mathbb{R}$ has subsets called intervals. An open interval $(a, b) = \{x : a < x < b\}$ is visualized on a number line as a segment between $a$ and $b$ with empty circles at the endpoints. A closed interval $[a, b] = \{x : a \le x \le b\}$ is visualized with solid/filled circles at the endpoints, indicating they are included.