CLASS XI | SETS

SETS — Complete Chapter

Structured, interactive explanations with step‑by‑step examples and MathJax rendering.

Tip: Click a topic bar to expand/collapse. All formulas render with MathJax; no extra setup needed.

Sets

Introduction, definition, notation

Introduction of sets

A set is a well‑defined collection of distinct objects (called elements), usually denoted by capital letters \(A,B,C,\dots\).

Example (well‑defined vs not well‑defined):

1. Claim: \(A=\{\text{vowels of English}\}\) is well‑defined.

\(A=\{a,e,i,o,u\}\) — membership is unambiguous.

2. Counter: \(B=\{\text{tall students in class}\}\) is not well‑defined unless “tall” is specified (e.g., height \(\ge 170\) cm).

Definition of sets

Elements are written inside curly braces. We use \(\in\) (is an element of) and \(\notin\) (is not an element of).

Example: Let \(S=\{2,4,6,8\}\).

1. \(4\in S\). 2. \(5\notin S\). 3. The number of elements (cardinality) is \(n(S)=4\).

Notation of sets

Common symbols: \(\emptyset\) (empty set), \(\subseteq\) (subset), \(\subset\) (proper subset), \(\cup\) (union), \(\cap\) (intersection), \(A'\) (complement w.r.t. universal set \(U\)).

Example: If \(U=\{1,2,3,4,5\}\) and \(A=\{2,4\}\), then the complement is

\(A' = U\setminus A = \{1,3,5\}\).

Representation

Set‑builder, roster, properties, conversion

Set builder form representation

Describe a set via a property: \(\{x \mid \text{condition on }x\}\).

Example: Multiples of 3 less than 12.

1. Define: \(A=\{x\in\mathbb{N}\mid 3\mid x,\ x<12\}\). 2. List to verify: \(\{3,6,9\}\).

Roster form representation

List elements explicitly inside braces.

Example: Divisors of \(12\).

1. Compute divisors: \(1,2,3,4,6,12\). 2. Roster: \(D=\{1,2,3,4,6,12\}\).

Properties of roster form (repetition, order)

• Repetition ignored: \(\{1,2,2,3\}=\{1,2,3\}\).
• Order irrelevant: \(\{1,2,3\}=\{3,2,1\}\).

Example: Show \(\{a,b,a\}=\{b,a\}\).

1. Remove duplicates → \(\{a,b\}\). 2. Order does not matter → equal sets.

Conversion of set builder form to roster form

Example: Convert \(S=\{x\in\mathbb{Z}\mid -2\le x<3\}\) to roster.

1. Integers from \(-2\) up to \(2\): \(-2,-1,0,1,2\). 2. So \(S=\{-2,-1,0,1,2\}\).

Types of set

Null, singleton, finite, infinite, equal

Null set

A set with no elements, denoted \(\emptyset\) or \(\{\}\).

Example: \(A=\{x\in\mathbb{N}\mid x^2=-1\}=\emptyset\).

1. For natural numbers, \(x^2\ge 0\). 2. So no solution → empty set.

Singleton set

A set with exactly one element.

Example: \(B=\{x\in\mathbb{Z}\mid x^2=9,\ x>0\}=\{3\}\).

1. Solutions to \(x^2=9\): \(\pm 3\). 2. With \(x>0\), we get \(x=3\) only → singleton.

Finite set

A set with finitely many elements.

Example: Vowels \(V=\{a,e,i,o,u\}\) ⇒ \(n(V)=5\) (finite).

Infinite set

A set with infinitely many elements.

Example: Natural numbers \(\mathbb{N}=\{1,2,3,\dots\}\) are infinite.

Equal set

Two sets are equal if they have exactly the same elements.

Example: \(A=\{1,2,3\},\ B=\{3,2,1\}\Rightarrow A=B\).

1. Show \(A\subseteq B\) and \(B\subseteq A\). 2. Hence equal.

Subset

Definition, representation, intervals, power, universal

Definition of subset

\(A\subseteq B\) means every element of \(A\) is also an element of \(B\).

Example: \(A=\{1,2\},\ B=\{1,2,3\}\Rightarrow A\subseteq B\).

1. Check \(1\in B\) and \(2\in B\). 2. All elements of \(A\) lie in \(B\) → subset.

Representation of subset

Use \(\subseteq\) (subset), \(\subset\) (proper subset), \(\supseteq\) (superset).

Example: If \(A=\{2,4\}, B=\{2,4,6\}\), then \(A\subset B\) and \(B\supset A\).

Subset of real numbers

Any set whose elements are real numbers is a subset of \(\mathbb{R}\).

Example: \(S=\{x\in\mathbb{R}\mid x^2=4\}=\{-2,2\}\subset\mathbb{R}\).

1. Solve \(x^2=4\) → \(x=\pm2\). 2. Both are real → subset of \(\mathbb{R}\).

Interval as subset of real numbers

Intervals: \((a,b),[a,b),(a,b],[a,b]\subset\mathbb{R}\).

Example: \([0,1)\subset\mathbb{R}\).

1. Contains all reals \(x\) with \(0\le x<1\). 2. \(1\notin [0,1)\); \(0,0.5\in [0,1)\).

Super set

If \(A\subseteq B\), then \(B\supseteq A\).

Example: \(B=\{1,2,3\}\) is a superset of \(A=\{2,3\}\) since \(A\subseteq B\).

Proper set (proper subset)

\(A\subset B\) means \(A\subseteq B\) and \(A\ne B\).

Example: \(\{1,2\}\subset \{1,2,3\}\) (proper subset).

Power set

The set of all subsets of \(A\), denoted \(P(A)\). If \(n(A)=k\), then \(n(P(A))=2^k\).

Example: \(A=\{a,b,c\}\).

1. Subsets: \(\emptyset,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\). 2. So \(P(A)\) has \(2^3=8\) elements.

Universal set

\(U\) contains all elements under discussion; complements are defined relative to \(U\).

Example: Let \(U=\{1,2,3,4,5,6\},\ A=\{2,4,6\}\).

1. \(A'=U\setminus A=\{1,3,5\}\). 2. \(A\cap A'=\emptyset,\ A\cup A'=U\).

Venn diagram

Definition, representation, use

Definition of Venn diagram

A rectangle represents \(U\); circles inside represent sets. Overlaps show intersections.

Example: Two sets \(A,B\).

1. Left circle \(A\), right circle \(B\) inside \(U\). 2. Overlap is \(A\cap B\); outside circles is \((A\cup B)'\).

Representation of Venn diagram

Shade regions to indicate operations like \(\cup,\cap,\setminus\).

Example: Shade \(A\setminus B\).

1. Take region in \(A\) not overlapping \(B\). 2. Symbolically \(A\setminus B = A\cap B'\).

Use of Venn diagram

Visualize counting problems and relationships between sets.

Example: If \(n(A)=20,\ n(B)=25,\ n(A\cap B)=5\), find \(n(A\cup B)\).

1. Formula: \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\). 2. Compute: \(20+25-5=40\).

Operation of sets

Union, intersection, complement, difference, symmetric difference, disjoint/exhaustive

Union of sets

\(A\cup B=\{x\mid x\in A \text{ or } x\in B\}\).

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

1. Combine unique elements. 2. \(A\cup B=\{1,2,3,4,5\}\).

Intersection of sets

\(A\cap B=\{x\mid x\in A \text{ and } x\in B\}\).

Example: With \(A,B\) above, \(A\cap B=\{2\}\).

Complement of sets

Relative to \(U\): \(A'=U\setminus A=\{x\in U\mid x\notin A\}\).

Example: \(U=\{1,2,3,4,5\},\ A=\{2,5\}\Rightarrow A'=\{1,3,4\}\).

Difference of sets

\(A\setminus B=\{x\mid x\in A,\ x\notin B\}\).

Example: \(A\setminus B=\{1,4\}\) for \(A=\{1,2,4\}, B=\{2,3,5\}\).

Symmetric difference of sets

\(A\triangle B=(A\setminus B)\cup(B\setminus A)\).

Example: For the same \(A,B\),

1. \(A\setminus B=\{1,4\}\), \(B\setminus A=\{3,5\}\). 2. \(A\triangle B=\{1,3,4,5\}\).

Disjoint sets

Sets with empty intersection: \(A\cap B=\emptyset\).

Example: \(A=\{1,3\},\ B=\{2,4\}\Rightarrow A\cap B=\emptyset\).

Mutually exclusive set

Another term for disjoint sets (especially in probability). No common outcomes.

Example: Rolling a die: \(A=\{\text{even}\},\ B=\{\text{odd}\}\) are mutually exclusive.

Mutually exhaustive sets

Sets whose union equals the universal set: \(A\cup B\cup\cdots=U\).

Example: On a die, even vs odd: \(A\cup B=U=\{1,2,3,4,5,6\}\) → exhaustive.

Mutually exclusive and exhaustive sets

A partition of \(U\): pairwise disjoint and union is \(U\).

Example: Die outcomes partitioned by remainder mod 3:

1. \(R0=\{3,6\},\ R1=\{1,4\},\ R_2=\{2,5\}\). 2. Disjoint and \(R0\cup R1\cup R_2=U\).

Properties of sets

Union, intersection, complement, commutative, associative, idempotent, distributive, De Morgan

Union properties

• Identity: \(A\cup\emptyset=A\).
• Domination: \(A\cup U=U\).
• Monotonicity: \(A\subseteq B\Rightarrow A\cup C\subseteq B\cup C\).

Example: With \(A=\{1\}, U=\{1,2\}\): \(A\cup\emptyset=\{1\}\), \(A\cup U=U\).

Intersection properties

• Identity: \(A\cap U=A\).
• Domination: \(A\cap\emptyset=\emptyset\).
• Monotonicity: \(A\subseteq B\Rightarrow A\cap C\subseteq B\cap C\).

Example: With \(A=\{1\}, U=\{1,2\}\): \(A\cup\emptyset=\{1\}\), \(A\cup U=U\).

Intersection properties

• Identity: \(A\cap U=A\).
• Domination: \(A\cap\emptyset=\emptyset\).
• Monotonicity: \(A\subseteq B\Rightarrow A\cap C\subseteq B\cap C\).

Example: \(A=\{1,2\}, U=\{1,2,3\}\Rightarrow A\cap U=A\).

Complement properties

• \((A')'=A\).
• \(A\cup A'=U\), \(A\cap A'=\emptyset\).

Example: \(U=\{1,2,3,4\}, A=\{1,4\}\Rightarrow A'=\{2,3\}\).

1. \(A\cup A'=\{1,2,3,4\}=U\). 2. \(A\cap A'=\emptyset\).

Commutative properties

\(A\cup B=B\cup A\) and \(A\cap B=B\cap A\).

Example: \(A=\{1,2\}, B=\{2,3\}\).

1. \(A\cup B=\{1,2,3\}=B\cup A\). 2. \(A\cap B=\{2\}=B\cap A\).

Associative properties

\((A\cup B)\cup C=A\cup(B\cup C)\) and \((A\cap B)\cap C=A\cap(B\cap C)\).

Example: \(A=\{1\}, B=\{1,2\}, C=\{2,3\}\).

1. \((A\cup B)\cup C=\{1,2\}\cup\{2,3\}=\{1,2,3\}\). 2. \(A\cup(B\cup C)=\{1\}\cup\{1,2,3\}=\{1,2,3\}\).

Idempotent properties

\(A\cup A=A\) and \(A\cap A=A\).

Example: Adding or intersecting a set with itself changes nothing: \( \{1,2\}\cup\{1,2\}=\{1,2\}\).

Distributive properties

\(A\cup(B\cap C)=(A\cup B)\cap(A\cup C)\) and \(A\cap(B\cup C)=(A\cap B)\cup(A\cap C)\).

Example: \(A=\{1,2\}, B=\{2,3\}, C=\{2,4\}\).

1. LHS: \(B\cap C=\{2\}\), so \(A\cup(B\cap C)=\{1,2\}\). 2. RHS: \(A\cup B=\{1,2,3\}, A\cup C=\{1,2,4\}\), intersection \(=\{1,2\}\). 3. LHS = RHS.

De Morgan's properties

\((A\cup B)'=A'\cap B'\) and \((A\cap B)'=A'\cup B'\).

Example: \(U=\{1,2,3,4,5\}, A=\{1,2\}, B=\{2,5\}\).

1. \(A\cup B=\{1,2,5\}\Rightarrow (A\cup B)'=\{3,4\}\). 2. \(A'=\{3,4,5\}, B'=\{1,3,4\}\Rightarrow A'\cap B'=\{3,4\}\). 3. Hence \((A\cup B)'=A'\cap B'\).

Formula of sets

n(A∪B), n(A−B), n(A′), Venn method for numericals

\(n(A\cup B)\)

\(n(A\cup B)=n(A)+n(B)-n(A\cap B)\).

Example: \(n(A)=30,\ n(B)=25,\ n(A\cap B)=7\).

1. Plug in: \(30+25-7=48\). 2. So \(n(A\cup B)=48\).

\(n(A-B)\)

\(n(A\setminus B)=n(A)-n(A\cap B)\).

Example: \(n(A)=30,\ n(A\cap B)=7\Rightarrow n(A\setminus B)=30-7=23\).

\(n(A')\)

With universal set \(U\): \(n(A')=n(U)-n(A)\).

Example: \(n(U)=60,\ n(A)=24\Rightarrow n(A')=60-24=36\).

Venn diagram method to solve numerical

Example (two sets word problem):

1. In a class of \(50\) students, \(n(M)=28\) like Math, \(n(S)=26\) like Science, and \(8\) like both. 2. Draw two overlapping circles \(M\) and \(S\). Put \(8\) in \(M\cap S\). 3. Only \(M\): \(28-8=20\). Only \(S\): \(26-8=18\). 4. Total who like at least one: \(20+8+18=46\). 5. Neither: \(50-46=4\).

Example (three sets quick setup):

1. Use regions: only \(A\), only \(B\), only \(C\), pairwise overlaps, and \(A\cap B\cap C\). 2. Fill from the deepest intersection outward to avoid double counting. 3. Apply identities like \(n(A)=\text{only }A+\text{pairs with }A+\text{all three}\).