Sets

A. What is a Set?

A Set is a well-defined collection of distinct objects:

• "Well-defined" Criterion: There must be an unambiguous, universal rule allowing anyone to decide definitively whether a specific object belongs to the collection or not.
  • Valid Set: "The vowels in the English alphabet" $\to$ Definite collection $\{a, e, i, o, u\}$.
  • NOT a Set: "The five most talented actors in India" $\to$ Subjective opinion; not well-defined!
• Elements / Members: The objects belonging to a set are called elements. If $x$ is an element of set $A$, we write $x \in A$ ("$x$ belongs to $A$"). If not, $x \notin A$.
• Elements are distinct (no duplicates) and the order of listing does not matter ($\{1, 2, 3\} = \\{3, 1, 2\}$).

B. The Three Methods of Representation:

1. 1. Description Method: Stating the well-defined rule in words inside quotes or text (e.g., $A = \text{Set of prime numbers less than 10}$).
2. 2. Roster / Tabular Form: Listing all elements explicitly, separated by commas, enclosed within curly braces $\{ \}$: $$A = \{2, 3, 5, 7\}$$
3. 3. Set-Builder / Rule Form: Stating the characteristic property $P(x)$ shared by all elements: $$A = \{x : x \text{ is a prime number and } x < 10\}$$ (Read as: "The set of all $x$ such that $x$ is a prime number less than 10").

A. Types of Sets:

• 1. Finite Set: A set containing a countable, finite number of elements (e.g., days of the week, $n(A) = 7$).
• 2. Infinite Set: A set containing an endless number of elements (e.g., $\mathbb{N} = \{1, 2, 3, 4, \dots\}$).
• 3. Empty / Null / Void Set ($\emptyset$ or $\{ \}$): A set containing zero elements (e.g., $\{x : x \in \mathbb{N} \text{ and } 2 < x < 3\} = \emptyset$). (Note: $\{0\}$ is NOT empty; it is a singleton set containing 0!).
• 4. Singleton Set: A set containing exactly one element (e.g., $\{0\}, \{5\}$).
• 5. Universal Set ($U$ or $\xi$): The overarching master set containing all objects under consideration for a specific problem.

B. Cardinality, Equivalent Sets vs Equal Sets:

• Cardinal Number $n(A)$: The total number of distinct elements in a finite set $A$.
• Equivalent Sets ($A \leftrightarrow B$): Two finite sets having the exact same number of elements: $n(A) = n(B)$. (e.g., $A = \{a, b, c\}$, $B = \{1, 2, 3\} \implies n(A) = n(B) = 3$).
• Equal Sets ($A = B$): Two sets containing the exact same identical elements, regardless of order. ($A = B \implies A \leftrightarrow B$, but equivalent sets are not necessarily equal!).

C. Subsets ($A \subseteq B$) & Power Sets:

• $A$ is a subset of $B$ ($A \subseteq B$) if every element of $A$ is also an element of $B$.
• Proper Subset ($A \subset B$): $A \subseteq B$ and $A \neq B$ (i.e., $B$ contains at least one element not in $A$).
• Two Universal Subset Axioms:
  1. The Empty Set is a subset of every set: $\emptyset \subseteq A$.
  2. Every set is a subset of itself: $A \subseteq A$.
• Total Subsets Formula: A set with $n$ elements has $2^n$ subsets and $2^n - 1$ proper subsets.

For any two sets $A$ and $B$ within universal set $U$:

1. Union of Sets ($A \cup B$):

The set of all elements that belong to $A$, or to $B$, or to both:

Example: $\{1, 2, 3\} \cup \{3, 4, 5\} = \{1, 2, 3, 4, 5\}$.

2. Intersection of Sets ($A \cap B$):

The set of elements common to both $A$ and $B$:

Example: $\{1, 2, 3\} \cap \{3, 4, 5\} = \{3\}$. If $A \cap B = \emptyset$, the sets are called Disjoint Sets.

3. Difference of Sets ($A - B$):

The set of elements that belong to $A$ but do NOT belong to $B$:

Example: $\{1, 2, 3, 4\} - \{3, 4, 5\} = \{1, 2\}$. (Note: $A - B \neq B - A$).

4. Complement of a Set ($A'$ or $A^c$):

The set of all elements in the universal set $U$ that do not belong to $A$:

A. Venn Diagrams (John Venn):

• The Universal Set ($U$) is represented by a large Rectangle.
• Subsets ($A, B$) are represented by enclosed circles drawn inside the rectangle. Overlapping areas represent the intersection $A \cap B$.

B. The Cardinality Addition Theorem:

For any two finite sets $A$ and $B$:

Why subtract $n(A \cap B)$? Because elements in the common intersection are counted twice—once in $n(A)$ and once in $n(B)$. Subtracting the intersection prevents double counting!

If $A$ and $B$ are disjoint sets ($A \cap B = \emptyset$), then: