Sets

2.1 Core concepts

Origin and motivation. The theory of sets was developed by Georg Cantor (1845–1918) while working on trigonometric series; the concept of a set is now a fundamental part of present-day mathematics and underlies relations, functions, geometry, sequences and probability (NCERT §1.1, p. 1). Cantor's revolutionary idea was to treat infinite collections as completed mathematical objects — a step that drew controversy in his lifetime but is now standard.

Definition of a set. A set is a well-defined collection of objects — meaning, given any object, we can decide unambiguously whether or not it belongs (NCERT §1.2, p. 2). Thus "the rivers of India" is a set, but "the five most renowned mathematicians of the world" is not, because the criterion is subjective. NCERT cites further well-defined collections: odd natural numbers less than 10, the rivers of India, vowels in the English alphabet, all even integers; and ill-defined collections like the most talented writers of India or honest persons of a country (p. 2).

Notation conventions. Objects, elements and members are synonyms; sets are denoted by capital letters (A, B, C, X, Y, Z) and elements by small letters (a, b, c, x, y, z). The symbol ∈ is read "belongs to" and ∉ "does not belong to" (NCERT §1.2, p. 2). For example if V is the set of vowels then a ∈ V but b ∉ V.

Standard number sets. N (natural numbers), Z (integers), Q (rationals), R (reals), Z⁺, Q⁺, R⁺ (their positive parts) carry fixed meanings throughout (NCERT §1.2, p. 2). The letter T is later used for the set of irrationals (p. 10). These names recur in every later chapter, so memorising them is non-negotiable.

Roster (tabular) form. All elements are listed inside braces { }, separated by commas; order is immaterial and elements are not repeated. For example, letters of "SCHOOL" → {S, C, H, O, L}. Dots are used to indicate an indefinite continuation: {1, 3, 5, …} for the set of all positive odd numbers (NCERT §1.2, p. 2–3).

Set-builder form. All elements share a common property not possessed by any element outside the set, written V = {x : x is a vowel in English alphabet}. The colon ":" reads "such that" and the braces read "the set of all" (NCERT §1.2, p. 3). NCERT often asks the student to translate between the two forms — e.g. {1, 4, 9, 16, 25, …} = {x : x = n², n ∈ N}.

Empty / null / void set. A set containing no element, denoted φ or { }. Examples: {x : 1 < x < 2, x ∈ N}, {x : x² – 2 = 0 and x is rational}, even primes greater than 2 — all are empty (NCERT §1.3, p. 5–6).

Finite vs infinite sets. A set is finite if it is empty or has a definite number of elements; otherwise it is infinite. n(S) denotes the number of distinct elements. All infinite sets cannot be written in roster form — e.g. the set of real numbers does not follow a pattern (NCERT §1.4, p. 6–7).

Equal sets. A = B iff every element of A is in B and every element of B is in A. Repetition is irrelevant: {1, 2, 3} = {2, 2, 1, 3, 3} (NCERT §1.5, p. 7–8).

Subsets and proper subsets. A ⊂ B if every element of A is also an element of B, i.e. a ∈ A ⇒ a ∈ B. Every set is a subset of itself; φ is a subset of every set. A ⊂ B and B ⊂ A ⇔ A = B (NCERT §1.6, p. 9). If A ⊂ B and A ≠ B, then A is a proper subset of B, and B is the superset of A (p. 10). A singleton set contains exactly one element, e.g. {a}.

Subsets of R. N ⊂ Z ⊂ Q, Q ⊂ R, T ⊂ R (T = irrationals), and N ⊄ T (NCERT §1.6.1, p. 10–11). Intervals: for a < b: open (a, b) = {y : a < y < b}; closed [a, b] = {x : a ≤ x ≤ b}; half-open [a, b), (a, b]. (–∞, ∞) is R. The length of any of these intervals is b – a (NCERT §1.6.2, p. 11–12).

Universal set, Venn diagrams. The universal set U is the basic set relevant to a context; all subsets in that context are taken from U (NCERT §1.7, p. 12). Venn diagrams, named after John Venn (1834–1883), depict U as a rectangle and its subsets as closed curves (usually circles).

Set operations. Union: A ∪ B = {x : x ∈ A or x ∈ B}; common elements counted once. If B ⊂ A then A ∪ B = A. Intersection: A ∩ B = {x : x ∈ A and x ∈ B}; if A ∩ B = φ, the sets are disjoint. Difference: A – B = {x : x ∈ A and x ∉ B}; in general A – B ≠ B – A. The three sets A – B, A ∩ B, B – A are mutually disjoint (NCERT §1.9, p. 14–17).

Complement. A′ = {x : x ∈ U and x ∉ A} = U – A; obviously A′ ⊂ U and (A′)′ = A. De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′ — the complement of a union is the intersection of complements, and vice versa (NCERT §1.10, p. 18–19).

Algebraic laws. Union and intersection are commutative and associative; ∩ distributes over ∪ and vice versa; identity laws A ∪ φ = A, U ∩ A = A; idempotent A ∪ A = A, A ∩ A = A; complement laws A ∪ A′ = U, A ∩ A′ = φ; double complementation (A′)′ = A; and φ′ = U, U′ = φ (NCERT §1.9.1–1.10, p. 14–20).

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

NCERT supplies a small but crucial gallery of Venn diagrams. Memorise the visual shape of each so you can re-create the answer instantly:

• Fig 1.1, p. 11 — Real-number line showing open, closed and half-open intervals as subsets of R. The open endpoint is an empty circle; the closed endpoint is a filled circle.
• Fig 1.2 and Fig 1.3, p. 13 — Venn diagram of universal set U = {1,…,10} with subset A = {2,4,6,8,10}, and a second showing B ⊂ A with B = {4, 6}. The proper-subset diagram is one circle drawn entirely inside another.
• Fig 1.4, p. 14 — Two overlapping circles with the entire shaded region representing A ∪ B. The union covers both lobes and the central lens.
• Fig 1.5, p. 15 — Two overlapping circles with only the central lens shaded — A ∩ B. This is the most common Venn shape on CUET answer keys.
• Fig 1.6, p. 15 — Two non-overlapping circles representing disjoint sets A and B (no lens, no overlap).
• Figs 1.7 (i)–(v), p. 16 — Five-step Venn-diagram proof of the distributive law A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). Students should trace through each shading step at least once.
• Fig 1.8, p. 17 — Shaded crescent representing A – B; only the part of A that lies outside B.
• Fig 1.9, p. 17 — Three mutually disjoint regions A – B, A ∩ B, B – A — together they partition A ∪ B.
• Fig 1.10, p. 19 — Universal-set rectangle with circle A; shaded region outside the circle (inside the rectangle) is A′. Process for finding n(A ∪ B ∪ C). Step 1 — write the addition formula n(A∪B∪C) = n(A) + n(B) + n(C) − n(A∩B) − n(B∩C) − n(C∩A) + n(A∩B∩C). Step 2 — substitute the given counts. Step 3 — solve for whichever quantity is unknown. This is the standard tool for the "students who read three newspapers" type problem. Process for converting roster ↔ set-builder. Look for a pattern: arithmetic progression → {x : x = a + (n−1)d}; squares → {x : x = n²}; multiples → {x : x = kn}. Conversely, plug consecutive n = 1, 2, 3 … into the rule to write the roster.

2.4 Common confusions / NTA trap points

• "Well-defined" vs "any collection". A collection like "ten most talented writers" is not a set because membership is subjective (NCERT §1.2, p. 2; Exercise 1.1 Q1).
• Element vs subset. For A = {1, 2, {3, 4}, 5}, the object {3, 4} is an element of A (so {3, 4} ∈ A) and {{3, 4}} is a subset of A — NCERT explicitly warns that an element should not be confused with the singleton containing it (Example 11 and Exercise 1.3 Q3, p. 10, 12).
• φ ⊂ A always; φ ∈ A only if φ is listed. The empty set is a subset of every set, but it is an element of a set only when written inside its braces (NCERT §1.6, p. 9; Exercise 1.3 Q3).
• A – B ≠ B – A. Difference is not commutative — students assume it behaves like ∪ or ∩ (NCERT §1.9.3, Example 18, p. 16).
• De Morgan flips the connective. (A ∪ B)′ becomes A′ ∩ B′, not A′ ∪ B′. NTA often offers both as distractors (NCERT §1.10, Example 22, p. 19).
• "Equal" requires same elements, not same count. {1, 2, 3} and {a, b, c} have the same n(S) but are not equal — only sets with identical elements are equal. They are however equivalent (NCERT §1.5, Definition 3, p. 7).
• Roster repetition. Letters of "FOLLOW" → {F, O, L, W}; repeated letters are listed once. Distractors keep duplicates to trap (NCERT §1.2 Note, p. 3; Exercise 1.2 Q5).
• Power-set count. |P(A)| = 2ⁿ, not n² — a frequent CUET swap. For A with 4 elements, P(A) has 2⁴ = 16 subsets, not 16 elements of A.
• Intervals with mixed endpoints. [a, b) includes a but excludes b. Re-read every option before committing.
• Inclusion–exclusion sign error. In n(A ∪ B) = n(A) + n(B) − n(A ∩ B), the intersection is subtracted, not added — a classic algebra slip.
• φ vs {φ}. φ has zero elements; {φ} has one element (namely φ). So |{φ}| = 1.

2.5 Key formulas & theorems

2.6 Solved examples

Example 1 — Power set count. Find the number of subsets and proper subsets of A = {1, 2, 3, 4, 5}.

Step 1 — |A| = 5. Step 2 — Number of subsets = 2⁵ = 32. Step 3 — Number of proper subsets = 2⁵ − 1 = 31 (excludes A itself). Answer: 32 subsets, 31 proper subsets.

Example 2 — Inclusion-exclusion (two sets). In a class of 100 students, 60 read English newspapers, 35 read Hindi newspapers, and 25 read both. How many read at least one?

Step 1 — n(E) = 60, n(H) = 35, n(E ∩ H) = 25. Step 2 — n(E ∪ H) = n(E) + n(H) − n(E ∩ H) = 60 + 35 − 25 = 70. Answer: 70 students read at least one newspaper.

Example 3 — Three-set survey. In a survey of 200 students: 80 like tea, 100 like coffee, 70 like milk; 35 tea+coffee, 30 coffee+milk, 20 tea+milk, 10 like all three. How many like none?

Step 1 — n(T ∪ C ∪ M) = 80 + 100 + 70 − 35 − 30 − 20 + 10 = 175. Step 2 — Students liking none = 200 − 175 = 25. Answer: 25 students like none.

Example 4 — Set algebra simplification. Simplify (A ∪ B) ∩ (A ∪ B′).

Step 1 — Distribute ∩ over ∪: A ∪ (B ∩ B′). Step 2 — Complement law: B ∩ B′ = φ. Step 3 — Identity: A ∪ φ = A. Answer: A.

Example 5 — De Morgan in computation. If U = {1,2,…,10}, A = {2,4,6,8,10}, B = {1,3,5,7,9}, verify (A ∪ B)′ = A′ ∩ B′.

Step 1 — A ∪ B = {1,2,…,10} = U, so (A ∪ B)′ = φ. Step 2 — A′ = {1,3,5,7,9} = B; B′ = {2,4,6,8,10} = A; A′ ∩ B′ = B ∩ A = φ. Answer: Both sides equal φ — verified.