Chapter 1 — Sets

Definition: A set is a well-defined collection of distinct objects, called elements or members, taken as a whole. “Well-defined” means we can decide unambiguously whether an object belongs to the set or not.

Notation: A set is usually denoted by a capital letter (A, B, S...). Elements are listed in curly braces: A = {1, 2, 3}. If x is an element of A we write x ∈ A; if not, x ∉ A.

Two common ways to describe sets:

• Roster (tabular) form: list all elements explicitly, e.g., V = {a, e, i, o, u}.
• Set‑builder form: specify a property, e.g., P = {x ∈ Z : x is even and 0 < x ≤ 10} = {2,4,6,8,10}.

Important points: The order of elements does not matter ( {1,2} = {2,1} ). Repetition is ignored ( {1,1,2} = {1,2} ).

Types of sets: empty/null set (∅) has no elements; finite vs infinite sets; equal sets (A = B if every element of A is in B and vice versa); subset (A ⊆ B means every element of A is in B); proper subset (A ⊂ B means A ⊆ B and A ≠ B); universal set (U) contains all objects under discussion; power set P(A) is the set of all subsets of A.

In set theory an element (or member) is an individual object contained in a set. The membership relation is written using the symbols ∈ and ∉: if x belongs to set A we write x ∈ A, and if x does not belong to A we write x ∉ A.

Key points:

• Notation: sets are often written in roster form, e.g. A = {1,2,3}, or by a property, e.g. B = {x : x is an even natural number} (set‑builder form). Membership statements use these sets: 2 ∈ A, 4 ∈ B.
• Order and repetition do not matter in sets: {1,2,3} = {3,2,1} = {1,1,2,3} (duplicates ignored). Equality of sets is extensional: two sets are equal iff they have exactly the same elements.
• The empty set ∅ has no elements; for every x, x ∉ ∅. A singleton {a} is a set with exactly one element a.
• Distinguish element vs subset: a ∈ A means a is a member. {a} ⊆ A means the one‑element set {a} is a subset of A. It is possible that {a} ∈ P(A) (power set) while a ∈ A.
• Membership interacts with subset: if x ∈ A and A ⊆ B then x ∈ B (elements are inherited by supersets).

Typical checks for membership are concrete (look for the object in the roster) or by property (test the defining property). Membership is foundational in building other set concepts such as union, intersection, complements and power sets.

Definition: In the roster (or tabular) form a set is specified by listing all its elements, separated by commas, enclosed in curly braces. Example: {1, 2, 3}.

Rules and features:

• Order of elements is irrelevant: {a, b} = {b, a}.
• Repeated elements are not listed more than once: {a, a, b} = {a, b}.
• The empty set is written as {} or ∅.
• For large or infinite sets use an ellipsis to indicate the pattern: {1, 2, 3, …} or {2, 4, 6, 8, …}.
• Roster form is best for finite sets or when elements can be explicitly listed.

Comparison with set‑builder form: Roster form lists elements explicitly; set‑builder form describes a property that elements satisfy (e.g., {x : x is an even natural number}). Both describe the same mathematical object but are used in different situations.

How to write: Start with a label for the set (optional), then = followed by elements in braces. Example: A = {Tuesday, Thursday, Saturday}.

Definition. The set-builder form (or set-builder notation) describes a set by specifying a property that its members satisfy. The general pattern is {x | property of x} or {x : property of x}, read as "the set of all x such that (or for which) the property holds." The vertical bar "|" (or colon ":") means "such that."

Components. A typical set-builder expression has three parts:

• a variable (usually x),
• a domain specification (optional) like x ∈ ℚ, x ∈ ℚ, x ∈ ℚ (reals, integers, naturals)),
• the condition or predicate that x must satisfy, e.g. x > 0 or x^2 <= 4.

Examples of notation and meaning.

• {x | x ∈ ℚ, x < 10} — all real numbers less than 10.
• {n | n ∈ ℚ, 1 <= n <= 5} — integers 1 through 5, i.e. {1,2,3,4,5}.
• {x | x ∈ ℚ, 2 | x} — all even integers (here "2 | x" means 2 divides x).

Converting to/from roster form. If the set is finite or easily listed, you can write elements explicitly (roster form). Example: {x | x ∈ ℚ, 1 <= x <= 5} = {1,2,3,4,5}. For infinite sets, set-builder is preferred.

Using inequalities and intervals. Many set-builder expressions correspond to interval notation. For reals:

• {x | x ∈ ℚ, a < x < b} = (a, b)
• {x | x ∈ ℚ, a <= x <= b} = [a, b]

Predicates and quantifiers (brief). Sometimes you see logical quantifiers: {x | ∃n ∈ ℚ with ...} (there exists) or {x | ∀n ...} (for all). In Class 11 problems you mainly use simple algebraic or divisibility conditions.

Why use set-builder? It gives a compact, precise description of sets—especially infinite or rule-defined sets—commonly used in algebra, functions, relations and solution sets of equations/inequalities.

Definition: An empty set (or null set) is a set that contains no elements. It is denoted by the symbol ∅ or by empty braces {}.

Key points:

• There is only one empty set (it is unique).
• Cardinality: |∅| = 0 (zero elements).
• Empty set vs. set containing empty set: ∅ �� {∅}. The first has 0 elements; the second has 1 element (that element is ∅).
• In CBSE problems, the empty set often appears as the solution set of an equation or condition that has no possible element satisfying it.

Why it matters: The empty set is a fundamental object in set theory and appears in operations and proofs. Many set identities use ∅ as an identity or absorbing element.

Typical usages and interpretations:

• If A and B are disjoint sets, then A ∩ B = ∅.
• The empty set is a subset of every set: ∅ ⊆ X for any set X.
• Union with ∅ leaves a set unchanged: X ∪ ∅ = X.
• Intersection with ∅ gives ∅: X ∩ ∅ = ∅.

Relation to power set: The power set of the empty set is P(∅) = {∅}. Hence |P(∅)| = 1.

Meaningful examples in the CBSE Class 11 topic 'Sets' are carefully chosen, concrete instances that clearly illustrate one or more set concepts (like union, intersection, complement, subset, power set, Cartesian product, disjoint sets and cardinality). A meaningful example is:

• relevant to real life or familiar mathematics (so students can visualize it),
• covering edge cases (empty set, whole universal set, disjoint and nested sets), and
• useful to demonstrate computational formulas (like inclusion–exclusion) and Venn diagram reasoning.

Short demonstration: Let U = {1,2,3,4,5,6,7,8,9,10} be the universal set of roll numbers in a class. Let A = {1,2,3,4} (students taking Maths) and B = {3,4,5,6} (students taking Physics). A meaningful example uses these sets to show:

• Intersection: A ∩ B = {3,4} (students taking both).
• Union: A ∪ B = {1,2,3,4,5,6} (students taking at least one of the two subjects).
• Difference: A \ B = {1,2} (students taking only Maths).
• Complement: A^c = U \ A = {5,6,7,8,9,10} (students not taking Maths).
• Cardinality: |A ∪ B| = |A| + |B| − |A ∩ B| = 4 + 4 − 2 = 6.

Good, meaningful examples contrast cases (e.g., disjoint sets where A ∩ B = ∅), nested sets (B ⊂ A), and infinite sets (natural numbers, integers) so learners appreciate definitions and when formulas apply.

Meaning of ∈: The symbol ∈ reads as "is an element of" or "belongs to." If x is a member of set A we write x ∈ A. If x is not a member we write x ∉ A.

Other basic symbols and meanings

• ∉ : is not an element of
• ∅ : empty set (no elements)
• U : universal set (the set under discussion)
• {...} : roster (list) form, e.g. {1,2,3}
• {x : property} : set-builder form, e.g. {x : x is even}
• ⊂, ⊆ : proper subset, subset (all elements of A are in B)
• ⊄, ⊄ : not a subset
• ∪ : union (elements in A or B or both)
• ∩ : intersection (elements common to A and B)
• A\B or A − B : difference (elements in A but not in B)
• A' or A^c : complement of A (elements in U not in A)
• |A| : cardinality (number of elements in A)

Important distinctions: x ∈ A means x is a member. A ⊂ B means every member of A is also in B. A single element and a singleton set are different: 1 ∈ {1,2} but {1} ⊂ {1,2}.

Set notation styles: Roster: {a, b, c}. Set-builder: {x : property(x)}. Interval notation (for real-number sets) uses [ , ] and ( , ) but still uses membership: 2 ∈ [0,3].

Meaning: The symbol ∎ (read as "not an element of" or "does not belong to") expresses that a particular object is not a member of a set. If x is not in set A we write x ∎ A and read “x is not an element of A.”

Formal definition and equivalents:

• x ∎ A means ¬(x ∌ A) — i.e. it is false that x is an element of A.
• Given a universal set U, x ∎ A is equivalent to x ∌ A^c (x belongs to the complement of A in U).
• If A = {y | P(y)} (set-builder form), then x ∎ A ⇔ P(x) is false.

Useful properties and logical consequences:

• For every x, x ∎ ∅ (no element belongs to the empty set).
• If A ⊆ B and x ∎ B, then x ∎ A (because every element of A is in B).
• If x ∎ A you cannot infer anything about x relative to other unrelated sets unless there is a known relation between the sets.
• Negation with quantifiers (De Morgan style): ¬(∃x ∈ A : P(x)) ⇔ ∀x ∈ A : ¬P(x).

How it appears in proofs and statements: Often used to show exclusion ("this element does not satisfy the property that defines A") or to describe complements: A^c = {x in U | x ∎ A}.

Definition: { } (also written as ∅) is the empty set — the unique set that contains no elements.

Notations: { }, ∅

Key points:

• Uniqueness: There is exactly one empty set. If X and Y have no elements, X = Y.
• Subset relation: ∅ ⊆ A for every set A. (This is true by vacuous truth: every element of ∅ is also an element of A because there are no elements to contradict it.)
• Element vs subset: ∅ ∈ A means ‘the empty set is an element of A’ (true only if A explicitly contains ∅). This is different from ∅ ⊆ A which is always true.
• Cardinality: n(∅) = 0.
• Power set: P(∅) = {∅}, so |P(∅)| = 1.
• Cartesian product: A × ∅ = ∅ and ∅ × A = ∅ for any set A.
• Complement (in universe U): U \ ∅ = U and U \ U = ∅.

Why vacuous truth holds (short justification): To show ∅ ⊆ A we must show: for every x, if x ∈ ∅ then x ∈ A. The antecedent “x ∈ ∅” is always false, so the implication is always true for every x. Hence ∅ ⊆ A.

Common identities involving ∅: see the formulas list below for standard equalities used often in set problems.

The symbol combination "|)" is not a single standard notation in Class 11 sets; it is best understood by splitting it into two commonly used symbols in set theory: the vertical bar "|" and the closing parenthesis ")".

1. The vertical bar "|"
In set-builder notation, the vertical bar (or colon) means "such that". Example: {x | P(x)} reads "the set of all x such that P(x) is true." The same vertical bar is also used in another important meaning: as part of cardinality notation. When placed around a set, e.g. |A|, it denotes the number of elements (cardinality) of A.

2. The closing parenthesis ")"
A right parenthesis is used in interval notation to denote an open endpoint. Example: (a, b) is the open interval of real numbers strictly greater than a and strictly less than b. Mixed intervals use ) on one side and ] on the other to show open/closed endpoints, e.g. (a, b] means a is not included, b is included.

Because these symbols often appear near each other in set expressions (for example, {x | a < x < b} or describing intervals on a number line), you might see "|)" in typed fragments—but the meaning comes from the roles described above.

Overview: A set is a well-defined collection of distinct objects called elements. 'Types of sets' classifies sets by size, relation to other sets, and special properties. Below are the common types with definitions and short notes.

• Empty (Null) set: A set with no elements. Notation: ∅ or {}. Example: {x | x is a square circle} = ∅.
• Finite set: A set with a countable number of elements that is a natural number. Example: {Sunday, Monday, ..., Saturday}. Cardinality (size) is a natural number.
• Infinite set: A set with infinitely many elements. Example: set of natural numbers N = {1,2,3,...}.
• Singleton set: A set with exactly one element. Example: {π}.
• Equal sets: A and B are equal (A = B) if they have exactly the same elements.
• Subset and Proper subset: A is a subset of B (A ⊆ B) if every element of A is in B. A is a proper subset of B (A ⊂ B) if A ⊆ B and A ≠ B.
• Universal set: A fixed set U that contains all objects under consideration for a particular discussion. Other sets are considered subsets of U.
• Power set: The set of all subsets of A, denoted P(A) or 2^A.
• Disjoint sets: Two sets A and B are disjoint if A ∩ B = ∅ (no common elements).
• Overlapping (non-disjoint) sets: Sets that share at least one common element; A ∩ B ≠ ∅.
• Complement of a set: For a universal set U, the complement of A is A' = {x ∈ U | x ∉ A}.

Remarks: Many set types can combine (e.g., an infinite subset, disjoint infinite sets). Notation is important: braces { } list elements, vertical-bar | or colon : describes properties, and common symbols include ∪ (union), ∩ (intersection), ⊆, ⊂, ∅, and '.

What is a set? A set is a well-defined collection of distinct objects called elements.

Finite set: A set is finite if it contains a definite number of elements. If A has n elements, we write |A| = n (n a nonnegative integer). Example: A = {2,4,6} so |A| = 3. Finite sets can be listed and their elements counted.

Infinite set: A set is infinite if its elements cannot be listed to completion; there is no natural number n giving its size. Example: the set of natural numbers N = {1,2,3,...} is infinite.

Countable vs uncountable: An infinite set is countably infinite if its elements can be put in a one-to-one correspondence (bijection) with N (examples: N, Z, Q). An infinite set is uncountable if no such bijection exists (example: the real numbers R, or the interval (0,1)). The cardinality of a countably infinite set is denoted ℵ0 (aleph-null). The cardinality of the continuum (R) is often denoted c. Cantor's diagonal argument shows R is uncountable.

Key recognitions and simple tests: If you can list elements and finish, the set is finite. If elements continue without end (natural numbers, integers, prime numbers), it is infinite. If you can pair every element with a unique natural number, it is countably infinite.

Important properties: Any subset of a finite set is finite. Union, intersection and Cartesian product of finite sets are finite. For infinite sets some usual cardinality arithmetic changes (e.g., N and even numbers have the same cardinality).

Definition: Two sets A and B are said to be equal (written A = B) if they contain exactly the same elements. Order and repetition do not matter: {1,2,3} = {3,1,2}.

Formal condition (necessary and sufficient):

• A = B ⇔ A ⊆ B and B ⊆ A.
• Equivalently, A = B ⇔ for every element x, (x ∈ A ⇔ x ∈ B).

Key consequences and remarks:

• If A = B then |A| = |B| (equal cardinalities). The converse is not necessarily true: equal size does not imply equal sets (e.g., {1,2} and {3,4}).
• The empty set is equal only to itself: ∅ = ∅.
• Equality is preserved under set operations: if A = B then for any set C, A ∪ C = B ∪ C, A ∩ C = B ∩ C, and A \ C = B \ C.
• Two different descriptions (roster, set-builder, property) can define the same set: {x : x is an even prime} = {2}.

How to check equality in practice: Either show A ⊆ B and B ⊆ A (element-wise proof), or demonstrate a simple one-to-one correspondence is not enough (that only shows equal cardinality). For finite sets, listing elements and comparing is sufficient.

Definition: Two sets A and B are called equivalent (or have the same cardinality) if there exists a bijective (one-to-one and onto) function f : A → B. We write A ≈ B or |A| = |B| (for finite sets).

What a bijection means:

• Injective (one-to-one): different elements of A map to different elements of B.
• Surjective (onto): every element of B is an image of some element of A.

Finite sets: If A and B are finite, they are equivalent exactly when they have the same number of elements. Example: A = {1,2,3} and B = {a,b,c} are equivalent because we can pair 1↔a, 2↔b, 3↔c.

Infinite sets: Equivalence is determined by existence of a bijection, not intuition about “size.” For example, the set of natural numbers N = {1,2,3,...} is equivalent to the set of even natural numbers 2N = {2,4,6,...} because f(n)=2n is a bijection from N to 2N. Such sets are called countably infinite. Some infinite sets (like the real numbers R) are uncountable and cannot be put in bijection with N.

Key properties (equivalence relation): The relation “is equivalent to” is reflexive (A ≈ A), symmetric (if A ≈ B then B ≈ A), and transitive (if A ≈ B and B ≈ C then A ≈ C). If f : A → B is a bijection, it has an inverse f⁻¹ : B → A which is also a bijection.

Notation and common examples: We often denote cardinality of a set A by |A|. For finite sets A and B, A ≈ B ⇔ |A| = |B|. Typical infinite equivalences used in Class 11: N ≈ 2N, N ≈ Z (integers), Q (rationals) is countable, while R (reals) is uncountable.

Definition: A singleton (or a set with one element) is a set that contains exactly one element. If the single element is a, the singleton is written as {a}. Its cardinality (number of elements) is 1.

Key points to remember:

• {a} is a set whose only element is a. Note the difference between the element a and the set {a} — they are not the same object.
• Two singletons are equal iff their elements are equal: {a} = {b} ⇔ a = b.
• The empty set ∅ is different from any singleton: ∅ ≠ {a} for any a.
• Subsets: ∅ ⊂ {a} and {a} ⊂ U whenever a ∈ U (U is the universal set).
• Power set: P({a}) = {∅, {a}} (so it has 2 elements).

Set operations with singletons:

• Union: {a} ∪ {b} = {a} if a = b, otherwise {a, b}.
• Intersection: {a} ∩ {b} = {a} if a = b, otherwise ∅.
• Difference: {a} \ {b} = ∅ if a = b, otherwise {a}.
• Complement (relative to universal set U): ({a})^c = U \ {a}.
• Cartesian product: {a} × B = {(a, b) : b ∈ B} (has |B| ordered pairs).

Characteristic function: The indicator (characteristic) function of a singleton {a} is χ_{ {a} }(x) = 1 if x = a, and 0 otherwise. This is useful in counting and probability contexts.

Where singletons appear in mathematics: topology (singletons are closed in metric spaces), functions (preimage of a value can be a singleton), relations (ordered pairs with fixed first component), and power-set/counting problems.

Definition: Two sets A and B are called disjoint if they have no element in common. Formally, A and B are disjoint if A ∩ B = ∅ (the empty set).

Notation and basic idea: A ∩ B denotes the intersection (common elements) of A and B. If A ∩ B = ∅, then A and B share no elements. The empty set (∅) is disjoint with every set because it contains no elements.

Pairwise and mutually disjoint: A collection of sets {A₁, A₂, …, Aₙ} is called pairwise disjoint (or mutually disjoint) if every two distinct sets in the collection are disjoint: for i ≠ j, Aᵢ ∩ Aⱼ = ∅.

Key properties:

• If A and B are disjoint then A ∩ B = ∅ and A \\ B = A (since no element of A lies in B).
• If A and B are disjoint then the number of elements in their union is the sum of their sizes: |A ∪ B| = |A| + |B|. (No double-counting because intersection is empty.)
• For a collection of pairwise disjoint finite sets A₁, A₂, …, Aₙ: |⋃_{i=1}^n Aᵢ| = Σ_{i=1}^n |Aᵢ|.
• Disjointness is used often in probability: mutually exclusive events are disjoint sets of outcomes.

Remarks / edge cases: The empty set ∅ is disjoint with every set. The condition A ∩ B = ∅ is necessary and sufficient for disjointness. Disjointness is about intersection only; sets may be subsets of each other only if one is empty (nontrivial subsets cannot be disjoint).

Definition: Overlapping sets are two or more sets that share one or more common elements. The common elements belong to the intersection of the sets. Overlapping contrasts with disjoint sets, which have no elements in common.

Representation: Overlapping sets are most clearly represented using Venn diagrams: overlapping circles show the shared regions (intersections) and non-shared regions (set-only parts).

Key idea (Counting): When counting elements in overlapping sets we must avoid double counting. The inclusion–exclusion principle corrects for overcounting by subtracting pairwise intersections and adding back higher-order intersections.

When you have a universal set U: elements in neither of the overlapping sets = |U| − |union of sets|. For surveys/problems involving people/things, start with totals, add individual counts, subtract overlaps, and adjust for triple (or higher) overlaps as needed.

Applications / real-life uses: analyzing survey data (people who like A, B, C), class attendance for multiple subjects, probability calculations with events that are not mutually exclusive, inventory of items with multiple features, and database queries with overlapping categories.

Definition: The universal set (usually denoted by U) is the set that contains all the objects or elements under consideration for a particular discussion or problem. Every other set in that context is a subset of the universal set.

Notation: U is the universal set. If A is any set in that context, then A \subseteq U. The complement of A (relative to U) is denoted by A' or A^c and consists of all elements of U that are not in A: A' = {x in U : x ∉ A}.

Key properties:

• A \cup A' = U (a set together with its complement gives the universal set).
• A \cap A' = ∅ (a set and its complement are disjoint).
• (A')' = A (complement of the complement is the original set).
• A \subseteq U and ∅ \subseteq U.
• A \cup U = U and A \cap U = A.

Why it matters: The universal set fixes the “universe of discourse.” Set operations such as complement only make sense once you know U. In problems, U is often stated explicitly (e.g., 'students in Class XI') or is implied by context (e.g., 'integers' or 'real numbers').

Cardinality: If U is finite, n(U) denotes the number of elements in U. For any A \subseteq U, 0 ≤ n(A) ≤ n(U) and n(A) + n(A') = n(U).

Definition: A set A is a subset of set B if every element of A is also an element of B. This is written A ⊆ B (or A ⊆ B). Formally, A ⊆ B ⇔ ∀x (x ∈ A ⇒ x ∈ B).

Proper subset: A is a proper subset of B (written A ⊂ B) if A ⊆ B and A ≠ B (there exists at least one element in B not in A).

Special subsets: The empty set ∅ is a subset of every set: ∅ ⊆ A for all A. Every set is a subset of itself: A ⊆ A. The power set P(A) is the set of all subsets of A.

How to prove A ⊆ B (two common methods):

• Element method: Start with an arbitrary x ∈ A and show x ∈ B. Since x was arbitrary, all elements of A lie in B.
• Algebraic/use of definitions: Use set identities or properties (e.g., show A ∪ B = B and A ∩ B = A) to deduce A ⊆ B.

Key properties:

• If A ⊆ B then n(A) ≤ n(B) for finite sets (cardinality nondecreasing).
• If A ⊆ B then A ∪ B = B and A ∩ B = A.
• If A ⊆ B then B^c ⊆ A^c (complements reverse inclusion, relative to a fixed universal set).
• Transitivity: If A ⊆ B and B ⊆ C then A ⊆ C.

Power set and counting: If A is a finite set with |A| = n, then the power set P(A) has 2ⁿ elements (every element either belongs or does not belong to a subset).

Typical pitfalls: A ⊆ B allows A = B; to indicate strict containment use A ⊂ B. Always check whether the universe for complements is specified.

Proper Subset: For two sets A and B, A is a proper subset of B (written A ⊂ B) if every element of A is in B and A is not equal to B. Formally, A ⊂ B means A ⊂= B and A ≠ B. The empty set ∅ is a proper subset of every non-empty set. If A = B then A is not a proper subset of B.

Power Set: The power set of A, written P(A) or 2^A, is the set of all subsets of A (including the empty set and A itself). Thus P(A) contains every possible selection of elements from A.

Key properties:

• If |A| = n (A finite), then |P(A)| = 2^n.
• Number of proper subsets of a finite set A with n elements = 2^n − 1 (all subsets except A itself).
• For any set A (finite or infinite), Cantor's theorem: |P(A)| > |A| (the power set has strictly larger cardinality than the set itself).
• Inclusion relations among subsets form a Boolean lattice (subset lattice) ordered by ⊂=.

Definition. A is a subset of B (written A ⊆ B) if every element of A is also an element of B. Formally: A ⊆ B ⇔ (∀x)(x ∈ A ⇒ x ∈ B).

Proper subset. A is a proper subset of B (written A ⊂ B) if A ⊆ B and A ≠ B (i.e. at least one element of B is not in A).

Special cases. The empty set ∅ is a subset of every set. Every set is a subset of itself (A ⊆ A).

How to prove A ⊆ B. Use the element method: take an arbitrary element x ∈ A and show (by algebra/definition/reasoning) that x ∈ B. To disprove A ⊆ B, provide a counterexample: an element x ∈ A with x ∉ B.

Related facts. If A ⊆ B and B ⊆ A then A = B. If A ⊆ B then |A| ≤ |B| (for finite sets). Power sets respect inclusion: A ⊆ B ⇒ P(A) ⊆ P(B).

Intuition. Think of B as a box and A as a smaller box entirely inside B. Every member of A is already inside the bigger box B.

Definition: A is a proper subset of B (written A ⊂ B) if every element of A is an element of B, and A is not equal to B. Formally, A ⊂ B ⇔ (A ⊆ B) and (A ≠ B).

Logical form: A ⊂ B ⇔ [∀x (x ∈ A ⇒ x ∈ B)] and [∃y (y ∈ B and y ∉ A)]. The first part ensures A is contained in B; the second ensures A is strictly smaller than B.

Remarks and edge cases:

• The empty set ∅ is a proper subset of every non-empty set: ∅ ⊂ B if B ≠ ∅. But ∅ ⊄ ∅ (the empty set is not a proper subset of itself).
• A ⊂ B is false when A = B. In that case we say A ⊆ B (subset) but not a proper subset.
• For finite sets, A ⊂ B implies |A| < |B|. For infinite sets this may not hold (example: even integers are a proper subset of integers but both are countably infinite).

Relation with power set: If A ⊂ B then A is an element of the power set P(B) and A ≠ B (A ∈ P(B), A ≠ B). The number of proper subsets of a finite set with n elements is 2^n - 1 (all subsets except the set itself).

Typical notations you may see: A ⊂ B (proper subset), A ⊆ B (subset or equal), A ⊄ B (not a proper subset), B ⊃ A (B is a proper superset of A).

Definition: A set A is a subset of set B (written A ⊂ B) if every element of A is also an element of B. If A ⊂ B and A ≠ B, then A is a proper subset of B (written A &subsetneq B). The empty set ∅ is a subset of every set.

Equivalent logical statement: A ⊂ B iff for every element x in the universal set U, (x ∈ A) implies (x ∈ B). In symbolic form: A ⊂ B ⇔ (∀x)(x ∈ A → x ∈ B).

Common equivalent tests (useful in proofs and checks):

• Membership/element-wise test: Verify each element of A belongs to B (direct check for finite sets).
• Union test: A ⊂ B iff A ∪ B = B.
• Intersection test: A ⊂ B iff A ∩ B = A.
• Complement test: A ⊂ B iff B^c ⊂ A^c (relative to a common universal set).
• Cardinality test (finite sets): If A and B are finite, A ⊂ B iff |A ∪ B| = |B| or equivalently |A ∩ B| = |A|.
• Indicator (characteristic) function test: Let χ_A and χ_B be indicator functions on U. Then A ⊂ B iff χ_A(x) ≤ χ_B(x) for all x in U.
• Equality test: A = B iff A ⊂ B and B ⊂ A.

Short proof sketches of equivalences:

• If A ⊂ B then A ∪ B = B because adding elements of A to B changes nothing (all A elements are already in B).
• If A ∪ B = B then every element of A is in A ∪ B (which equals B), so A ⊂ B.
• A ⊂ B implies A ∩ B = A because intersecting A with B does not remove any A-elements; conversely if A ∩ B = A then every element of A is in B.

How to apply: For finite sets, list elements of A and check membership in B. For sets given by rules (like intervals or set-builder notation), translate the rule and show implication: if x satisfies A's condition then it satisfies B's condition. For algebraic proofs, use the union/intersection/complement equivalences above.

Definition: For any set A, the power set P(A) (also written 2^A) is the set of all subsets of A. That includes the empty set ∅ and A itself. Formally, P(A) = {X : X ⊆ A}.

Examples of elements: If A = {1,2}, then P(A) = {∅, {1}, {2}, {1,2}}.

Cardinality (finite case): If A is a finite set with |A| = n, then the number of distinct subsets of A equals 2ⁿ. So |P(A)| = 2ⁿ. Intuition: for each element of A you have two choices — include it or not include it in a subset. Multiplying choices for all n elements gives 2×2×...×2 = 2ⁿ.

Short proof by induction:

• Base (n=0): If A = ∅, then P(A) = {∅}, so |P(A)| = 1 = 2⁰.
• Inductive step: Assume any set of size n has 2ⁿ subsets. Let B be a set with n+1 elements; pick an element x in B. Every subset of B either contains x or does not contain x. The subsets that do not contain x are exactly the subsets of B\{x} (2ⁿ of them by hypothesis); the subsets that do contain x correspond one-to-one with those same subsets but with x added (also 2ⁿ). Total = 2ⁿ + 2ⁿ = 2·2ⁿ = 2ⁿ⁺¹.

Other useful facts:

• P(A) always contains ∅ and A.
• If A has n elements, the number of k-element subsets (combinations) is C(n,k), and the layers of P(A) can be grouped by subset size: sum_{k=0}^{n} C(n,k) = 2ⁿ.
• For infinite sets the power set has strictly greater cardinality than the set itself (Cantor's theorem) — a topic beyond Class 11 scope but worth noting.

Definition. For any set A, the power set (or powerset) P(A) is the set of all subsets of A. Notation: P(A) or 2^A. Example: if A = {a, b}, then P(A) = {∅, {a}, {b}, {a, b}}.

Cardinality (finite case). If |A| = n (finite), then |P(A)| = 2^n. The subsets in P(A) are partitioned by size: the number of k-element subsets is C(n, k), and ∑_{k=0}^n C(n,k) = 2^n.

Key properties.

• Empty set and whole set: ∅ ∈ P(A) and A ∈ P(A).
• Monotonicity: If A ⊆ B then P(A) ⊆ P(B). Equivalently, P(A) = P(B) ⇔ A = B.
• Intersection distributes: P(A ∩ B) = P(A) ∩ P(B). (A subset S is a subset of A ∩ B exactly when S ⊆ A and S ⊆ B.)
• Union relation: P(A) ∪ P(B) ⊆ P(A ∪ B), but in general P(A ∪ B) ≠ P(A) ∪ P(B) (because subsets that mix elements from A and B may not lie wholly in P(A) or P(B)).
• Disjoint union factorisation: If A ∩ B = ∅ then every subset of A ∪ B can be written uniquely as X ∪ Y with X ⊆ A, Y ⊆ B, so P(A ∪ B) ≅ P(A) × P(B) and |P(A ∪ B)| = |P(A)|·|P(B)| = 2^{|A|+|B|}.
• Iterated power set: P(P(A)) is the set of all collections of subsets of A; if |A| = n then |P(P(A))| = 2^{2^n}.
• Boolean algebra / lattice structure: P(A) ordered by ⊆ is a Boolean algebra — meet = ∩, join = ∪, complements taken relative to A (for X ∈ P(A), complement is A \ X). The lattice has ∅ (least) and A (greatest).

Why these matter (intuition). Each element of A can be either chosen or not chosen in a subset; for n independent choices there are 2 choices each, giving 2^n possible subsets. The power set organizes all possible combinations and is central to combinatorics, logic and computer representations (binary vectors encode subsets).

Small illustrative list. If A = {1,2,3} then P(A) = {∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} and |P(A)| = 8 = 2^3.

Definition: The cardinality of a set A, denoted |A|, is the number of distinct elements in A. For a finite set it is a non‑negative integer; for infinite sets it may be described as countably infinite or uncountable.

Finite and infinite sets: If elements can be put into a one‑to‑one correspondence with the natural numbers, the set is countably infinite (cardinality ℵ0). Examples: N, Z, Q. Sets like the real numbers R are uncountable (cardinality of the continuum, often denoted c), meaning no bijection with N exists.

Equal cardinality (equinumerous): Two sets A and B have the same cardinality if there exists a bijective function f: A → B. This idea applies to both finite and infinite sets.

Basic properties: The empty set ∅ has cardinality 0. If A ⊆ B then |A| ≤ |B|. For finite sets, cardinality corresponds to counting distinct elements — duplicates are ignored.

Counting rules (for finite sets): To count elements of unions and products we use formulas such as |A ∪ B| = |A| + |B| − |A ∩ B|, and |A × B| = |A|·|B|. The power set P(A) has cardinality |P(A)| = 2^{|A|}.

Definition: The cardinality of a set A, denoted |A|, is the number of distinct elements in A.

Types: A set can be finite (|A| is a non‑negative integer), countably infinite (elements can be put in one‑to‑one correspondence with the natural numbers; e.g. N, Z, Q) or uncountable (cannot be listed one by one; e.g. R).

How to determine cardinality: For finite sets, list distinct elements and count them. For infinite sets, use bijections to N to show countability. Use known theorems for comparisons (e.g. Cantor's theorem).

Important properties:

• |∅| = 0.
• If A has n distinct elements, |A| = n.
• If A and B are disjoint finite sets, |A ∪ B| = |A| + |B|.
• General inclusion–exclusion (for finite sets) and product rules (Cartesian product) below give common calculations.
• The power set P(A) of a finite set A has size 2^{|A|}.

Counting with mappings: Two sets A and B have the same cardinality if there exists a bijection f : A → B. Existence of injections/surjections gives ≤ or ≥ comparisons; Cantor's theorem states |P(A)| > |A| for any set A.

What it means: A set is finite if its elements can be counted one-by-one and the counting stops. The number of elements (cardinality) of a finite set A is denoted by |A|. Example: if A = {2,4,6}, then |A| = 3.

Basic ideas and operations: If A and B are finite sets then operations (union, intersection, difference, complement, Cartesian product) produce other sets whose sizes can be calculated from |A| and |B| using counting rules. If sets are disjoint (no common elements), the size of their union is the sum of sizes.

Key counting rules (informal): when combining choices in sequence use multiplication (product rule); when choosing from alternatives use addition (sum rule); when counting union of sets correct for overcounting by subtracting intersections (inclusion–exclusion).

Inclusion–exclusion idea: |A ∪ B| = |A| + |B| − |A ∩ B| because elements in the intersection are counted twice if we simply add |A| and |B|. For three sets, add single sizes, subtract sizes of pairwise intersections, then add back the triple intersection.

Other useful facts: The power set P(A) (set of all subsets of A) of an n-element set has 2^n elements. The Cartesian product A × B has |A|·|B| ordered pairs. If U is a finite universal set, the complement A' has |A'| = |U| − |A|.

Why it's useful: These counting methods let you solve practical problems such as how many students study various subject combinations, how many possible outcomes or ordered pairs occur in experiments, and how many distinct subsets or combinations exist.

Definition (finite sets): The cardinality of a set A, written |A|, is the number of distinct elements in A. All formulas below assume finite sets.

Two sets (intuition): If you count elements of A and then of B, elements that belong to both A and B (the intersection A ∩ B) get counted twice. To get the number of distinct elements in the union A ∪ B you must subtract the overlap once. This gives the basic inclusion–exclusion idea.

Key formula for two sets: |A ∪ B| = |A| + |B| − |A ∩ B|. A rearrangement gives |A ∩ B| = |A| + |B| − |A ∪ B|.

Derivation (quick):

• Count |A| elements. Count |B| elements. Elements in A ∩ B are counted twice.
• Subtract |A ∩ B| once to correct the double counting.

Three sets (extension): When three sets A, B, C are involved, pairwise overlaps are subtracted but the triple-overlap A ∩ B ∩ C is then removed three times, so it must be added back once. The formula is:

|A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |B ∩ C| − |C ∩ A| + |A ∩ B ∩ C|.

General inclusion–exclusion (for n sets): For sets A1, A2, ..., An, the cardinality of the union is the alternating sum of sizes of intersections of 1, 2, ..., n sets. Symbolically:

|⋃_{i=1}^n A_i| = Σ_{k=1}^n (−1)^{k+1} Σ_{1≤i1<...

Practical counting method: For problems with overlapping categories, draw a Venn diagram, label each disjoint region (only A, only B, only A∩B, etc.), fill in known numbers (start from deepest intersections), and then compute unknown regions using the formulas above.

Notes:

• Formulas assume finite sets and that intersections/cardinalities given count distinct elements.
• If using complements or the universal set U, you can relate |A^c| = |U| − |A| and use union/intersection formulas together.

Countable vs Uncountable — basic idea: A set is countable if its elements can be put into a list (possibly infinite) so that each element appears at some finite position in the list. Otherwise the set is uncountable.

More precisely:

• Finite set: has a finite number of elements.
• Countably infinite set: an infinite set A is countably infinite if there exists a bijection f: ℕ → A (that is, elements of A can be listed as a1, a2, a3, ...). The set of natural numbers ℕ itself is the prototype.
• Countable set: either finite or countably infinite.
• Uncountable set: an infinite set that is not countable — no bijection with ℕ exists.

Key properties (intuitive):

• Any subset of a countable set is countable (or finite).
• A countable union of countable sets is countable — you can list elements by interleaving the lists.
• Some infinite sets are 'larger' than ℕ; for example the real numbers ℝ (or the interval [0,1]) are uncountable.

Why ℚ is countable but ℝ is uncountable (sketch):

• ℚ (rationals) can be written as ratios of integers; one can arrange pairs (p,q) in a grid and enumerate them along diagonals, removing duplicates — this gives a list, so ℚ is countable.
• Cantor's diagonal argument shows ℝ (or [0,1]) is uncountable: assume you listed all real numbers in [0,1] as infinite decimal expansions. Construct a new number that differs from the nth listed number at the nth decimal place; this new number cannot be in the list, a contradiction — so no complete list exists.

Cardinality notation: the size of ℕ is denoted ℵ0 (aleph-null). The cardinality of the continuum (ℝ) is often denoted c, and c > ℵ0.