Sets and Functions - Sets, their representations, and types

Definition of a Set: A set is a well-defined collection of distinct objects, called elements or members. To be 'well-defined', there must be a clear rule to determine if an object belongs to the set or not. Visually, a set is often represented by a closed loop (Venn diagram) where the elements are points plotted inside the boundary.

Representation of Sets: Sets are expressed in two primary ways. The Roster (Tabular) form lists all elements separated by commas within braces, like $A = \{1, 2, 3\}$. The Set-Builder form describes the common property of the elements using a variable, such as $A = \{x : x \text{ is a natural number and } x < 4\}$. Visually, Roster form looks like a discrete list, while Set-Builder form looks like a mathematical condition or logical constraint.

Types of Sets based on Cardinality: An Empty Set (or Null Set) contains no elements, denoted by $\phi$ or $\{\}$. A Finite Set contains a countable number of elements, whereas an Infinite Set has elements that cannot be counted (e.g., the set of all stars). Visually, a finite set has a clear end to its list of elements, while an infinite set is denoted by an ellipsis '...' indicating it continues forever.

Equal vs. Equivalent Sets: Two sets $A$ and $B$ are Equal ($A = B$) if they have exactly the same elements. They are Equivalent ($A \approx B$) if they have the same number of elements (cardinality), meaning $n(A) = n(B)$. Visually, equal sets would be represented by the same circle in a Venn diagram, while equivalent sets might be different circles but containing the same number of distinct points.

Subsets and Proper Subsets: A set $A$ is a subset of $B$ ($A \subseteq B$) if every element of $A$ is also an element of $B$. If $A \subseteq B$ and $A \neq B$, then $A$ is a Proper Subset ($A \subset B$). Visually, this is shown as a smaller circle $A$ completely enclosed within a larger circle $B$.

Power Set: The collection of all possible subsets of a set $A$ is called the Power Set, denoted by $P(A)$. If a set has $n$ elements, the power set will contain $2^n$ elements. Visually, if you think of a set as a box of items, the power set represents every possible combination of items you could take out of that box, including taking nothing (the empty set) and taking everything.

Universal Set and Complement: The Universal Set ($U$) is the set containing all objects under consideration for a particular discussion. The complement of set $A$, denoted $A'$ or $A^c$, consists of all elements in $U$ that are not in $A$. Visually, if $U$ is a large rectangle and $A$ is a circle inside it, $A'$ is represented by the entire shaded area inside the rectangle but outside the circle.