Comprehensive Study Notes on Set Notes on Set Theory

Introduction to the Theory of Sets

• Definition of a Set: A set is a well-defined collection of objects. This means that for any given object, it is possible to definitely decide whether it belongs to the collection or not.
• Historical Development: The theory of sets was developed by the German mathematician Georg Cantor (1845-1918). He first encountered the concept while working on "problems on trigonometric series."
• Importance in Mathematics: Today, the concept of a set serves as a fundamental part of modern mathematics. It is used to define relations and functions and is essential in the study of geometry, sequences, and probability.
• Examples of Well-Defined Collections:
  • Odd natural numbers less than 10: {$1, 3, 5, 7, 9$}.
  • The vowels in the English alphabet: {$a, e, i, o, u$}.
  • Solutions of the equation $x^2 - 5x + 6 = 0$: {$2, 3$}.
  • Prime factors of 210: {$2, 3, 5, 7$}.
• Examples of Collections That Are Not Well-Defined: The collection of the "five most renowned mathematicians of the world" is not a set because the criterion for being "most renowned" varies from person to person.

Basic Notation and Special Number Sets

• Synonyms: The terms "objects," "elements," and "members" of a set are synonymous.
• Naming Conventions:
  • Sets are usually denoted by capital letters: $A, B, C, X, Y, Z$, etc.
  • Elements of a set are represented by small letters: $a, b, c, x, y, z$, etc.
• Membership Notation:
  • If $a$ is an element of set $A$, we write $a \in A$ (the Greek symbol epsilon $\in$ denotes "belongs to").
  • If $b$ is not an element of set $A$, we write $b \notin A$ ("does not belong to").
• Standard Symbols for Mathematical Sets:
  • $N$: The set of all natural numbers.
  • $Z$: The set of all integers.
  • $Q$: The set of all rational numbers.
  • $R$: The set of real numbers.
  • $Z^+$: The set of positive integers.
  • $Q^+$: The set of positive rational numbers.
  • $R^+$: The set of positive real numbers.
  • $T$: The set of irrational numbers (defined as the set of real numbers that are not rational, i.e., $x \in R$ and $x \notin Q$).

Methods of Representing Sets

• Roster or Tabular Form:
  • All elements are listed, separated by commas, and enclosed within braces ${ }$.
  • Rule of Order: The order in which elements are listed is immaterial. For example, {$1, 2, 3$} is the same as {$3, 2, 1$}.
  • Rule of Repetition: Elements are generally not repeated; they are taken as distinct. For example, the set of letters in the word "SCHOOL" is {$S, C, H, O, L$}.
  • Infinite Roster: Infinite sets can be indicated by three dots. For example, the set of odd natural numbers is {$1, 3, 5, ...$}.
• Set-Builder Form:
  • All elements possess a single common property not shared by elements outside the set.
  • Notation: $V = \{x : x \text{ is a vowel in English alphabet}\}$. The braces stand for "the set of all," the variable $x$ represents the element, the colon ":" stands for "such that," and the description follows.
  • Example from Textbook: Writing the set $A = \{1, 4, 9, 16, 25, ...\}$ in set-builder form:
    • Alternative 1: $A = \{x : x \text{ is the square of a natural number}\}$
    • Alternative 2: $A = \{x : x = n^2, \text{ where } n \in N\}$
  • Example with Constraints: The set {$\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}$} is written as:
    • $\{x : x = \frac{n}{n+1}, \text{ where } n \text{ is a natural number and } 1 \le n \le 6\}$.

Types of Sets: Empty, Finite, and Infinite

• The Empty Set: Also called the null set or void set. It contains no elements at all.
  • Symbol: Denoted by $\phi$ or ${ }$
  • Examples:
    • $\{x : 1 < x < 2, x \in N\}$ (no natural number exists between 1 and 2).
    • $\{x : x^2 - 2 = 0 \text{ and } x \text{ is a rational number}\}$ (the roots are irrational).
    • The set of even prime numbers greater than 2.
• Finite Sets: A set that is empty or consists of a definite number of elements.
  • $n(S)$: Denotes the number of distinct elements in a finite set $S$.
• Infinite Sets: A set where the count of elements is not finite (e.g., the set of all points on a line or the set of natural numbers).
  • Note: Not all infinite sets can be represented in roster form. For example, the set of real numbers $R$ cannot be written in roster form because the elements do not follow a specific pattern that can be shown with dots.

Equality and Subsets

• Equal Sets: Two sets $A$ and $B$ are equal ($A = B$) if they have exactly the same elements. Repetition or order does not change the set.
• Subsets: Set $A$ is a subset of set $B$ ($A \subset B$) if every element of $A$ is also an element of $B$.
  • Logic: $A \subset B$ if $a \in A \Rightarrow a \in B$ (where $\Rightarrow$ means "implies").
  • Reflexivity: Every set is a subset of itself ($A \subset A$).
  • The Empty Set Rule: The empty set $\phi$ is a subset of every set.
  • Equality Condition: $A = B$ if and only if $A \subset B$ and $B \subset A$. This is often written as "iff" and symbolized by $\Leftrightarrow$.
• Proper Subsets and Supersets: If $A \subset B$ and $A \ne B$, then $A$ is called a proper subset and $B$ is called a superset of $A$.
• Singleton Set: A set that contains only one element, e.g., {$a$}.
• Real Number Subset Chains: $N \subset Z \subset Q \subset R$, $T \subset R$, and $N \not\subset T$.

Intervals as Subsets of R

• Let $a, b \in R$ and $a < b$:
  • Open Interval: $(a, b) = \{y : a < y < b\}$. End points are excluded.
  • Closed Interval: $[a, b] = \{x : a \le x \le b\}$. End points are included.
  • Semi-Open/Semi-Closed:
    • $[a, b) = \{x : a \le x < b\}$ (includes $a$, excludes $b$).
    • $(a, b] = \{x : a < x \le b\}$ (excludes $a$, includes $b$).
• Length of Interval: For any of the above intervals, the length is defined as $(b - a)$.
• Infinite Intervals:
  • $[0, \infty)$ defines non-negative real numbers.
  • $(-\infty, 0)$ defines negative real numbers.
  • $(-\infty, \infty)$ defines the entire set of real numbers.

Universal Set and Venn Diagrams

• Universal Set ($U$): The basic set relevant to a particular context from which all other sets are drawn.
  • In number theory, $U$ might be $R$. In human studies, $U$ might be the world population.
• Venn Diagrams: Graphical representations of sets, named after John Venn (1834-1883).
  • The universal set is usually represented by a rectangle.
  • Subsets are represented by closed curves (usually circles) within the rectangle.

Operations on Sets

• Union of Sets ($\cup$): The union of $A$ and $B$ is the set of all elements belonging to either $A$ or $B$ (or both).
  • $A \cup B = \{x : x \in A \text{ or } x \in B\}$.
  • Properties:
    • Commutative Law: $A \cup B = B \cup A$
    • Associative Law: $(A \cup B) \cup C = A \cup (B \cup C)$
    • Law of Identity: $A \cup \phi = A$ (where $\phi$ is the identity of union).
    • Idempotent Law: $A \cup A = A$
    • Law of U: $U \cup A = U$
• Intersection of Sets ($\cap$): The intersection of $A$ and $B$ is the set of all elements common to both $A$ and $B$.
  • $A \cap B = \{x : x \in A \text{ and } x \in B\}$.
  • Disjoint Sets: If $A \cap B = \phi$.
  • Properties:
    • Commutative Law: $A \cap B = B \cap A$
    • Associative Law: $(A \cap B) \cap C = A \cap (B \cap C)$
    • Law of $\phi$ and U: $\phi \cap A = \phi$, $U \cap A = A$
    • Idempotent Law: $A \cap A = A$
    • Distributive Law: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$. (Intersection distributes over union).
• Difference of Sets: The set of elements that belong to $A$ but not to $B$.
  • $A - B = \{x : x \in A \text{ and } x \notin B\}$.
  • Note: Generally, $A - B \ne B - A$.
  • Disjoint Relationships: The sets $(A - B)$, $(A \cap B)$, and $(B - A)$ are mutually disjoint.

Complement of a Set

• Definition: The complement of set $A$ (denoted as $A'$) is the set of all elements in the universal set $U$ that are not in $A$.
  • $A' = \{x : x \in U \text{ and } x \notin A\}$. Equivalently, $A' = U - A$.
• Complement Laws:
  • $A \cup A' = U$
  • $A \cap A' = \phi$
• De Morgan's Laws: Named after the mathematician De Morgan.
  • $(A \cup B)' = A' \cap B'$ (The complement of the union is the intersection of the complements).
  • $(A \cap B)' = A' \cup B'$ (The complement of the intersection is the union of the complements).
• Law of Double Complementation: $(A')' = A$.
• Laws of Empty and Universal Sets: $\phi' = U$, $U' = \phi$.

Historical Note and Theoretical Developments

• Pioneers: Georg Cantor originated set theory (1874-1897). His work was supported by Richard Dedekind (1831-1916) but criticized by Kronecker (1810-1893) regarding infinite sets.
• Russell's Paradox: In 1902, the English philosopher Bertrand Russell (1872-1970) showed that the assumption of the "set of all sets" leads to a contradiction. This led to the conclusion that "nothing contains everything" (as written by Paul R. Halmos in Naïve Set Theory).
• Axiomatization: To resolve paradoxes, set theory was axiomatized by several logicians:
  • Ernst Zermelo (1908).
  • Abraham Fraenkel (1922).
  • John Von Neumann (1925): Introduced the axiom of regularity.
  • Paul Bernays (1937): Provided a more satisfactory axiomatization.
  • Kurt G\u00f6del (1940): Modified axioms, resulting in the Von Neumann-Bernays (VNB) or G\u00f6del-Bernays (GB) set theory.
• Legacy: Despite initial resistance, set-theoretic language is now used to express almost all concepts in modern mathematics.