Key Points

A set is a well-defined collection of distinct objects. 'Well-defined' means it is possible to definitively determine if an object belongs to the collection.

Sets are represented in two main ways: Roster Form lists all elements, e.g., \{a, e, i, o, u\}, while Set-Builder Form describes elements by a common property, e.g., \{x : x \text{ is a vowel in the English alphabet}\}.

The empty set is a set containing no elements at all. It is denoted by the symbol \phi or by empty braces \{\}.

A set is finite if it is empty or consists of a definite number of elements. Otherwise, the set is called infinite, such as the set of natural numbers \{1, 2, 3, ...\}.

Two sets A and B are said to be equal if they have exactly the same elements, regardless of order or repetition. If sets A and B are equal, we write $A = B$.

A set A is a subset of set B, written $A \subset B$, if every element of A is also an element of B. If $A \subset B$ and $A \neq B$, then A is a proper subset of B.

Every set is a subset of itself ($A \subset A$). The empty set \phi is a subset of every set. If $A \subset B$ and $B \subset A$, then $A = B$.

Intervals are subsets of the set of real numbers R. An open interval (a, b) represents \{x : a < x < b\}, while a closed interval [a, b] represents \{x : a \leq x \leq b\}.

The Universal Set, denoted by U, is a basic set that contains all elements and subsets relevant to a particular discussion. All other sets in the context are considered subsets of U.

The union of two sets A and B, denoted $A \cup B$, is the set of all elements that are in set A, or in set B, or in both. Symbolically, $A \cup B = \{x : x \in A \text{ or } x \in B\}.

The intersection of two sets A and B, denoted $A \cap B$, is the set of all elements that are common to both A and B. Symbolically, $A \cap B = \{x : x \in A \text{ and } x \in B\}.

Two sets A and B are called disjoint if they have no elements in common. This means their intersection is the empty set, i.e., $A \cap B = \phi$.

The difference of sets A and B, denoted $A - B$, is the set of elements that belong to A but not to B. It is important to note that $A - B \neq B - A$ in general.

The complement of a set A, denoted $A'$, is the set of all elements in the universal set U that are not in A. It is defined as $A' = U - A$.

De Morgan's Laws relate the operations of union, intersection, and complement. The laws are: (i) $(A \cup B)' = A' \cap B'$ and (ii) $(A \cap B)' = A' \cup B'.

Union and Intersection are commutative ($A \cup B = B \cup A$) and associative ($(A \cup B) \cup C = A \cup (B \cup C)$). Intersection distributes over union: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$.