Comprehensive Study Notes on Set Theory for JEE Mathematics

Fundamental Definitions and Rules of Set Theory

• Definition of a Set: A set is a well-defined collection of objects. The term "well-defined" implies there must be no ambiguity regarding which objects are included or excluded from the group.
• Examples of Sets:   - Set of Fruits in a Basket.   - Set of Coins in a Bank.   - Set of Books in a Library.   - Set of Consonants in the English Alphabet.   - Collection of past Presidents of India.
• Non-Set Examples:   - Difficult topics in Mathematics.   - Group of Intelligent Students in a JEE Batch.
• The Adjective Rule: If a sentence describes a collection using adjectives such as "good," "difficult," "intelligent," "brave," or "smart," the collection does not qualify as a set because these terms are subjective and not well-defined.
• Notation Standards:   - Sets are usually denoted by capital letters: $A, B, C, \dots, X, Y, Z$.   - Individual members of a set are called elements or things.   - Elements are typically denoted by small letters: $a, b, c, \dots, x, y, z$.
• Fundamental Properties:   - Order: The order in which elements are written inside the curly brackets makes no difference (${a, b} = {b, a}$).   - Repetition: Elements are not allowed to repeat within a set. Every element must be distinct.

Notations and Standard Sets of Numbers

• Set of Natural Numbers ($N$): $\dots {1, 2, 3, \dots}$.
• Set of All Integers ($Z$ or $I$): $\dots {0, \pm 1, \pm 2, \dots}$.
• Set of Non-Zero Integers ($Z_0$ or $I_0$): $\dots {\pm 1, \pm 2, \pm 3, \dots}$.
• Set of All Rational Numbers ($Q$): Defined as ${x : x = \frac{p}{q}, \dots \text{where } p \text{ and } q \text{ are relatively prime integers and } q \neq 0}$.
• Set of Real Numbers ($R$): The set containing all rational and irrational numbers.
• Set of Complex Numbers ($C$): The broadest set, containing all numbers in the form $a + bi$.

Representation of Sets

• Tabulation Method (Roster Form):   - All elements are listed within curly brackets and separated by commas.   - Example: Set $A$ of days in a week: $A = {\text{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}}$.
• Set Builder Method (Set Rule Method):   - Instead of listing elements, a property or rule is used that is satisfied by all elements in the set.   - Example: Under the same weekly context, $A = {x : x \text{ is a day of the week}}$.

Cardinality and Basic Types of Sets

• Cardinal Number: The number of distinct elements in a finite set $A$, denoted by $n(A)$.
• Equivalent Sets: Two finite sets $A$ and $B$ are equivalent if they have the same cardinal number ($n(A) = n(B)$).
• Null Set (Void Set or Empty Set):   - A set having no elements at all.   - Denoted by $\phi$ or $\dots { }$.   - The null set is technically a subset of every existing set.   - Example: \dots {x : x \in N, 5 < x < 6} = \phi, as no natural number exists between 5 and 6.
• Finite Set: A set that is either empty or contains a definite (countable) number of elements.
• Infinite Set: A set whose elements cannot be counted (e.g., the set of all real numbers).
• Singleton Set (Unit Set):   - A set containing exactly one and only one element.   - Example: Positive integral roots of $x^2 - 2x - 15 = 0$. Factoring gives $(x + 3)(x - 5) = 0$, so $x = -3$ or $x = 5$. The set of positive integral roots is simply $\dots {5}$.
• Equal Sets: Two finite sets $A$ and $B$ that possess the exact same members. If $A = B$, then their cardinalities are identical, and every element of $A$ is in $B$ and vice versa.

Subsets, Supersets, and Power Sets

• Subset ($A \subseteq B$): Every element of set $A$ is also an element of set $B$.
• Superset: If $A \subseteq B$, then $B$ is the superset of $A$.
• Proper Subset ($A \subset B$): Occurs when $A \subseteq B$ but $A \neq B$ (set $B$ contains at least one element not in $A$).
• Subset Rules:   - Every set is a subset of itself ($A \subseteq A$).   - The empty set $\phi$ is a subset of every set.   - If $A \subseteq B$ and $B \subseteq A$, then $A = B$.
• Counting Subsets: If a set $A$ contains $n$ elements, it has $2^n$ total subsets.
• Power Set ($P(A)$): The collection of all possible subsets of set $A$.
• Formulas for a set with $m$ elements:   - Number of elements in the power set $n(P(A)) = 2^m$.   - Number of non-void (non-empty) subsets = $2^m - 1$.   - Number of proper subsets = $2^m - 1$.   - Number of non-void proper subsets = $2^m - 2$.
• Universal Set ($U$): The superset that contains all sets under consideration in a specific context (e.g., the set of complex numbers is the universal set for all number-related sets).

Mathematical Intervals as Subsets of Real Numbers

For $a, b \in R$ where a < b:

1. Open Interval $(a, b)$ or $]a, b[$: \dots {x : a < x < b}. Includes all numbers between $a$ and $b$, excluding the endpoints.
2. Closed Interval $[a, b]$: $\dots {x : a \leq x \leq b}$. Includes all numbers between $a$ and $b$ plus the endpoints.
3. Open-Closed Interval $(a, b]$ or $]a, b]$: \dots {x : a < x \leq b}. Endpoint $a$ is excluded, but $b$ is included.
4. Closed-Open Interval $[a, b)$ or $[a, b[$: \dots {x : a \leq x < b}. Endpoint $a$ is included, but $b$ is excluded.
5. Infinite Representations:    - x > a translates to $(a, \infty)$.    - $x \leq b$ translates to $(-\infty, b]$.    - Positive Real Numbers ($R^+$) = $(0, \infty)$.    - Negative Real Numbers ($R^-$) = $(-\infty, 0)$.    - All Real Numbers ($R$) = $(-\infty, \infty)$.

Operations on Sets

• Union ($A \cup B$): The set of elements belonging to $A$, or $B$, or both. Also known as the "Logical Sum."
• Intersection ($A \cap B$): The set of elements that belong simultaneously to both $A$ and $B$.
• Disjoint Sets: Two sets are disjoint if they have no common elements, resulting in $A \cap B = \phi$.
• Difference of Sets ($A - B$): All elements that belong to set $A$ but do not belong to set $B$. Defined as $\dots {x : x \in A \text{ and } x \notin B}$.   - Property: $A - B = A \cap B'$   - Property: $B - A = B \cap A'$   - Property: $(A - B) \cap B = \phi$
• Complement of a Set ($A'$ or $A^c$): The set of all elements in the Universal Set $U$ that are not in $A$. Defined as $\dots {x : x \in U \text{ and } x \notin A}$.   - Property: $A \cap A' = \phi$   - Property: $A \cup A' = U$   - Property: $(A')' = A$   - Property: $U' = \phi$ and $\phi' = U$
• Symmetric Difference ($A \Delta B$): The union of the differences of the sets: $(A - B) \cup (B - A)$. It contains elements that are in exactly one of the sets but not in their intersection.   - Formula: $A \Delta B = (A \cup B) - (A \cap B)$.
• Venn Diagrams: Successive pictorial representations where the Universal Set is a rectangle and subsets are circles inside it.

Laws of Algebra of Sets

• Idempotent Laws:   - $A \cup A = A$   - $A \cap A = A$
• Identity Laws:   - $A \cup \phi = A$   - $A \cap U = A$
• Commutative Laws:   - $A \cup B = B \cup A$   - $A \cap B = B \cap A$
• Associative Laws:   - $(A \cup B) \cup C = A \cup (B \cup C)$   - $(A \cap B) \cap C = A \cap (B \cap C)$
• Distributive Laws:   - $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$   - $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
• De Morgan's Laws:   - $(A \cup B)' = A' \cap B'$   - $(A \cap B)' = A' \cup B'$

The Cartesian Product

• Definition: The Cartesian product of sets $A$ and $B$, denoted $A \times B$, is the set of all ordered pairs $(a, b)$ such that $a \in A$ and $b \in B$.
• Cardinality: If set $A$ has $m$ elements and set $B$ has $n$ elements, then $A \times B$ has $mn$ elements.
• Example: If $A = {1, 2, 3}$ and $B = {a, b}$, then $A \times B = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}$, with $n(A \times B) = 3 \times 2 = 6$.
• Properties:   - $A \times B = \phi$ if either $A$ or $B$ is empty.   - $A \times (B \cup C) = (A \times B) \cup (A \times C)$.   - $A \times (B \cap C) = (A \times B) \cap (A \times C)$.   - $(A \times B) \cap (C \times D) = (A \cap C) \times (B \cap D)$.   - $A \times B = B \times A$ only if $A = B$.

Key Results on Number of Elements

• Two Sets:   - $n(A \cup B) = n(A) + n(B) - n(A \cap B)$.   - If $A$ and $B$ are disjoint, $n(A \cup B) = n(A) + n(B)$.
• Three Sets:   - $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$.
• Specific Sub-regions:   - Elements in exactly two of $A, B, C$: $n(A \cap B) + n(B \cap C) + n(C \cap A) - 3n(A \cap B \cap C)$.   - Elements in exactly one of $A, B, C$: $n(A) + n(B) + n(C) - 2(n(A \cap B) + n(B \cap C) + n(A \cap C)) + 3n(A \cap B \cap C)$.
• Complements:   - $n(A' \cup B') = n(U) - n(A \cap B)$.   - $n(A' \cap B') = n(U) - n(A \cup B)$.

Problem-Solving Scenarios and Case Studies

• Multiple Intersection Example: If $N_a = {ax : x \in N}$, find $3N \cap 7N$. Elements must be multiples of 3 AND multiples of 7. The common multiples start at the least common multiple ($LCM$), which is 21. Thus, $3N \cap 7N = 21N$.
• Newspaper Advertisement Case: City population word problem.   - Reads paper A: 25% ($n(L) = 25$).   - Reads paper B: 20% ($n(M) = 20$).   - Reads both: 8% ($n(L \cap M) = 8$).   - Readers of ONLY A: $25 - 8 = 17$.   - Readers of ONLY B: $20 - 8 = 12$.   - Advertisement reach: (30% of readers of Only A) + (40% of readers of Only B) + (50% of readers of Both) = $(0.3 \times 17) + (0.4 \times 12) + (0.5 \times 8) = 5.1 + 4.8 + 4 = 13.9\%$.
• Survey on Four Sets (Vacation Seasons):   - Given data for Summer ($S_u$), Winter ($S_w$), Spring ($S_p$), and Autumn ($S_a$) with various intersections.   - For a population of 100 employees, using the Principle of Inclusion-Exclusion for four sets, the calculation for employees taking vacations in EVERY season results in $n(S_u \cap S_p \cap S_w \cap S_a) = 2$.
• Consumer Survey (Product A and B):   - 1000 consumers surveyed. 720 like A, 450 like B.   - To find the least number who like both: $n(A \cap B)_{min} = n(A) + n(B) - n(U) = 720 + 450 - 1000 = 1170 - 1000 = 170$.
• Extreme Values in Union: If $n(A) = 3$ and $n(B) = 6$:   - Maximum $n(A \cup B)$ occurs when sets are disjoint ($3 + 6 = 9$).   - Minimum $n(A \cup B)$ occurs when $A \subseteq B$ ($n = 6$).

Questions & Discussion

• Question: Is the set of intelligent students a set?
• Answer: No, "intelligent" is an adjective and not well-defined, leading to ambiguity.
• Question: What is the result of $n(P(P(P(P(P(\phi))))))$?   1. $n(P(\phi)) = 2^0 = 1$   2. $n(P(P(\phi))) = 2^1 = 2$   3. $n(P(P(P(\phi)))) = 2^2 = 4$   4. $n(P(P(P(P(\phi))))) = 2^4 = 16$   5. $n(P(P(P(P(P(\phi)))))) = 2^{16} = 65536$.
• Question: If $A = {x : x \neq x}$, what does it represent?
• Answer: No object is not equal to itself, so the set has no elements. It is the empty set $\phi$ or $\dots { }$.