Sets revision notes

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: $A = \{x : x \text{ is a vowel in English alphabet}\}$.

• 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.

• 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 $\{2, 2, 1, 3, 3\}$.

• 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).

• 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 $N \not\subset T$.

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 $b$.

• 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.

• 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 $U - A$.

• Distributive Law: Intersection distributes over Union: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$.

• 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.

• Difference is Non-Commutative: Generally, $A - B \neq B - A$. Taking away elements is directional.
• Elements vs. Subsets ($\in$ vs $\subset$): An element belongs to ($\in$) a set, but a subset is contained in ($\subset$) a set. JEE Tip Trap: An element of a set can never be a subset of itself. If $A = \{1\}$ and $B = \{\{1\}, 2\}$, then $A \in B$. However, $A \not\subset B$.
• Mutually Disjoint Set Components: The sets $(A - B)$, $(A \cap B)$, and $(B - A)$ are mutually disjoint sets. The intersection of any two of these sets will always be the null set $\phi$.
• Roster Form Exclusions: All infinite sets cannot be described in roster form. Specifically, the set of Real Numbers ($R$) cannot be written this way because its elements do not follow a particular pattern.

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.

Power Sets

• Definition: The collection of all possible subsets of a set A is called the Power Set of A, denoted by $P(A)$.

• 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 $A \Delta B$.

• 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: