Sets revision notes

Key Concepts & Definitions

Set:

A well-defined collection of objects, meaning it is possible to definitively decide whether a given object belongs to the collection or not. For example, "odd natural numbers less than 10" is well-defined, but "the five most renowned mathematicians" is not subjective.

Elements:

The objects, elements, and members of a set are synonymous terms.

Notations:

Sets are conventionally denoted by capital letters (A, B, C, etc.), while their elements are represented by small letters (a, b, c, x, y, z, etc.).

Belongingness (∈\in∈):

If aaa is an element of set A, we say "aaa belongs to A" and write a∈Aa \in Aa∈A. If it does not belong, we write a∉Aa \notin Aa∈/A.

Logical Implication (  ⟹  \implies⟹):

The symbol   ⟹  \implies⟹ means "implies". For example, a∈A  ⟹  a∈Ba \in A \implies a \in Ba∈A⟹a∈B means "if aaa is an element of A, it implies that aaa is also an element of B".

Equivalence (  ⟺  \iff⟺):

The symbol   ⟺  \iff⟺ stands for two-way implications, usually read as "if and only if" (briefly written as "iff").

Representation of Sets

Sets can be represented mathematically using two primary methods:

1. Roster or Tabular Form:
   • All elements are listed, separated by commas, and enclosed within braces {}.
   • Rules: The order in which elements are listed is absolutely immaterial (e.g., $\{1, 2, 3\} = \{3, 1, 2\}$). Elements are generally not repeated; each element is taken as distinct (e.g., the set of letters in "SCHOOL" is $\{S, C, H, O, L\}$).
   • For infinite sets with a pattern, dots are used to signify continuation indefinitely (e.g., $\{1, 3, 5, ...\}$).
2. Set-Builder Form:
   • All elements possess a single common property not possessed by any element outside the set.
   • Described using a variable symbol (like $x$), followed by a colon $:$ (read as "such that"), then the characteristic property, and enclosed in braces.
   • Example: $A = \{x : x \text{ is a vowel in English alphabet}\}$.

Types of Sets

• Empty / Null / Void Set ($\phi$ or $\{\}$): A set which does not contain any element. → [JEE TIP] The set $\{\phi\}$ is not an empty set; it is a singleton set containing the empty set as an element.
• Finite and Infinite Sets:
  • Finite Set: A set that is empty or consists of a definite, countable number of distinct elements. The number of distinct elements in a finite set S is denoted by $n(S)$.
  • Infinite Set: A set whose elements cannot be counted, meaning the number of elements is not finite.
• Equal Sets: Two sets A and B are equal ($A = B$) if they have exactly the same elements. → [JEE TIP] A set does not change if one or more elements are repeated; $\{1, 2, 3\}$ is equal to $\{2, 2, 1, 3, 3\}$.
• Subset ($A \subset B$): A set A is a subset of B if every element of A is also an element of B ($a \in A \implies a \in B$).
  • The empty set $\phi$ is a subset of every set.
  • Every set is a subset of itself ($A \subset A$).
• Proper Subset and Superset: If $A \subset B$ and $A \neq B$, then A is a proper subset of B, and B is a superset of A.
• Singleton Set: A set containing exactly one element.
• Universal Set ($U$): The basic set containing all possible elements and subsets relevant to a particular context (e.g., the set of Real numbers when discussing subsets of integers).

Standard Mathematical Sets & Hierarchy

• N: Set of all natural numbers.
• Z: Set of all integers.
• Q: Set of all rational numbers ($Q = \{x : x = \frac{p}{q}, p, q \in Z \text{ and } q \neq 0\}$).
• R: Set of all real numbers.
• Z+, Q+, R+: Sets of positive integers, positive rational numbers, and positive real numbers respectively.
• T: Set of irrational numbers ($T = \{x : x \in R \text{ and } x \notin Q\}$).
• Hierarchy relations: $N \subset Z \subset Q \subset R$, and $T \subset R$. Note that $N \not\subset T$.

Intervals as Subsets of R

Let $a, b \in R$ and $a < b$:

• Open Interval $(a, b)$: $\{y : a < y < b\}$. All points between $a$ and $b$ belong to the interval, but $a$ and $b$ themselves are excluded.
• Closed Interval $[a, b]$: $\{x : a \le x \le b\}$. Contains all points between $a$ and $b$, including the endpoints $a$ and $b$.
• Semi-Open/Closed Intervals:
  • $[a, b)$: $\{x : a \le x < b\}$. Includes $a$ but excludes $b$.
  • $(a, b]$: $\{x : a < x \le b\}$. Excludes $a$ but includes $b$.
• Length of Interval: The length of any of these intervals is mathematically defined as $(b - a)$.
• Infinity: The set $[0, \infty)$ defines non-negative real numbers, $(-\infty, 0)$ defines negative real numbers, and $(-\infty, \infty)$ defines the entire real number line.

Operations on Sets

• Union of Sets ($A \cup B$): The set consisting of all elements that are either in A, in B, or in both. Symbolically: $A \cup B = \{x : x \in A \text{ or } x \in B\}$.
• Intersection of Sets ($A \cap B$): The set of all elements which are common to both A and B. Symbolically: $A \cap B = \{x : x \in A \text{ and } x \in B\}$.
• Disjoint Sets: If $A \cap B = \phi$ (no common elements), then sets A and B are disjoint.
• Difference of Sets ($A - B$): The set of elements which belong to A but not to B. Symbolically: $A - B = \{x : x \in A \text{ and } x \notin B\}$.
• Complement of a Set ($A'$): For a universal set U and $A \subset U$, the complement of A is the set of all elements of U that are not in A. Symbolically: $A' = \{x : x \in U \text{ and } x \notin A\}$. This is also strictly equal to $U - A$.

Formulae, Equations & Laws

• Properties of Union:
  • Commutative Law: $A \cup B = B \cup A$.
  • Associative Law: $(A \cup B) \cup C = A \cup (B \cup C)$.
  • Law of Identity Element: $A \cup \phi = A$ ($\phi$ is the identity of $\cup$).
  • Idempotent Law: $A \cup A = A$.
  • Law of U: $U \cup A = U$.
• Properties of Intersection:
  • 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: $\phi \cap A = \phi$ and $U \cap A = A$.
  • Idempotent Law: $A \cap A = A$.
• Distributive Law: Intersection distributes over Union: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$.
• Properties of Complement Sets:
  • Complement Laws: $A \cup A' = U$ and $A \cap A' = \phi$.
  • De Morgan's Laws:
    • $(A \cup B)' = A' \cap B'$ (The complement of union is the intersection of complements).
    • $(A \cap B)' = A' \cup B'$ (The complement of intersection is the union of complements).
  • Law of Double Complementation: $(A')' = A$.
  • Laws of Empty Set and Universal Set: $\phi' = U$ and $U' = \phi$.

Important Graphs & Diagrams (Venn Diagrams)

• Concept: Invented by John Venn, these represent sets using geometric shapes. The Universal set ($U$) is represented by a rectangle, and subsets by closed curves (usually circles). Elements are written inside their respective circles.
• Union ($A \cup B$): The entire region covered by both circles A and B.
• Intersection ($A \cap B$): Only the overlapping region between circles A and B.
• Disjoint Sets: Circles A and B do not overlap at all inside the universal rectangle.
• Difference ($A - B$): Only the region of circle A that does not overlap with circle B.
• Complement ($A'$): The entire area of the universal rectangle outside of circle A is shaded.

Historical Note

The modern theory of sets was largely originated by German mathematician Georg Cantor (1845-1918) while he was studying trigonometric series. He famously showed that the set of real numbers cannot be put into one-to-one correspondence with integers. Set theory encountered logical paradoxes (like Russell's Paradox in 1902, showing "nothing contains everything"). This later led to strict axiomatisation by mathematicians like Zermelo (1908), Fraenkel (1922), and Von Neumann.

⚠️ ADDITIONAL JEE CONCEPTS ⚠️

Power Sets

• Definition: The collection of all possible subsets of a set A is called the Power Set of A, denoted by $P(A)$.
• Cardinality: If a set A has $n$ elements (i.e., $n(A) = m$), then the number of elements in its power set is $2^m$.
  • Formula: $n(P(A)) = 2^m$
• → [JEE TIP] Every element of a power set is itself a set. The empty set $\phi$ and the set A itself are always elements of $P(A)$.

Symmetric Difference of Two Sets

• Definition: The symmetric difference of sets A and B is the set of elements which belong to exactly one of A or B, but not both. It is denoted by $A \Delta B$.
• Formulae:
  • $A \Delta B = (A - B) \cup (B - A)$
  • $A \Delta B = (A \cup B) - (A \cap B)$
• → [JEE TIP] In a Venn diagram, this represents the regions of A and B exclusively, leaving the intersection region completely blank.

Cardinality Theorems (Principle of Inclusion-Exclusion)

For any finite sets A, B, and C:

• Two Sets Formula:
  • $n(A \cup B) = n(A) + n(B) - n(A \cap B)$
  • If A and B are disjoint, then $n(A \cap B) = 0$, so $n(A \cup B) = n(A) + n(B)$.
• Three Sets Formula:
  • $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$
• Exact Number Properties:
  • Number of elements in exactly one of the sets A or B: $n(A \Delta B) = n(A) + n(B) - 2n(A \cap B)$
  • Number of elements in only A: $n(A - B) = n(A) - n(A \cap B)$