Sets
Well-defined collections, roster & set-builder forms, intervals, power sets, Venn diagrams & De Morgan's laws.
Topic 1.2: What is a Set? Roster Form vs Set-Builder Form
If your friend asks for a list of "Top 5 tastiest snacks in Kerala", every student in class will write a DIFFERENT list because "tasty" is a matter of personal opinion! But if you ask for "vowels in the English alphabet", EVERYONE writes
{a, e, i, o, u}. A collection with zero confusion — where everybody agrees YES or NO — is a Set!DEFINITIONA well-defined collection of objects (no opinion words like "best", "tall", "clever")
TWO FORMSRoster: list inside
{ }, no repeats. Set-Builder: rule {x : P(x)}SYMBOLS
∈ = "belongs to" | ∉ = "does not belong to"🧠 Memory Trick: Repeat letters vanish inside a set! Writing
SCHOOL in Roster Form becomes {S, C, H, O, L} — just 5 elements, order doesn't matter!💡 Simple Kid-Check: Think of a set like a labelled school lunchbox
A = {idli, sambar, chutney}. You never pack two identical names in the checklist, and whether you list idli first or chutney first, it's the exact same set!- 1Standard Number Sets:
ℕ(1, 2, 3...),ℤ(...−1, 0, 1...),ℚ(p/q fractions),ℝ(all real numbers). - 2Roster (Tabular) Form: List distinct elements separated by commas inside curly braces, e.g.
{2, 4, 6, 8}. - 3Set-Builder Form: State the common secret rule inside braces:
{x : x is an even natural number less than 10}.
Watch out for Kerala HSE 1-mark questions asking "Is the collection of most talented writers a set?" — Answer is NO, because "most talented" is subjective, NOT well-defined! Also remember: repeating elements in
MATHEMATICS gives only 8 letters {M, A, T, H, E, I, C, S}.Topic 1.3 – 1.5: Empty Set, Finite / Infinite Sets & Equal Sets
Imagine opening a cage marked "Live Dinosaurs in Kochi" — how many are inside? Zero! That empty cage is the Empty Set (∅ or { })! Meanwhile, all students in your school make a Finite Set (counting ends!), while all whole numbers
1, 2, 3, ... keep going forever as an Infinite Set!EMPTY SET (∅)Has 0 elements: written
∅ or {} (NEVER {∅} or {0}!)FINITE vs INFINITECount finishes → Finite; uncountable end (like
ℕ, ℝ) → InfiniteEQUAL SETS (A = B)Exactly same elements, even if repeated or reordered!
🧠 Memory Trick: Watch
{0} vs { }! A lunchbox containing the number zero {0} is NOT empty — it holds 1 item! Only { } or bare ∅ has zero items!💡 Simple Kid-Check: Empty set has zero items. A singleton set has 1 item (like
{2}, the only even prime!). Two sets are Equal (A = B) if they contain identical distinct members: {1, 2, 3} = {3, 1, 2, 2}.
🔎 Quick Examples:
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{x ∈ ℝ : x² + 1 = 0} = ∅ (no real number squares to −1!)•
{x ∈ ℝ : 1 < x < 2} is INFINITE (infinitely many decimals like 1.1, 1.01, 1.001 sit between 1 and 2!)
Exam Trap:
{∅} is NOT an empty set! It is a singleton box holding the symbol ∅ inside it, so n({∅}) = 1 while n(∅) = 0! Also remember {x ∈ ℤ : 1 < x < 2} = ∅ (finite empty), whereas {x ∈ ℝ : 1 < x < 2} is infinite!Topic 1.6: Subsets, Real Intervals & Power Set (2ⁿ)
Imagine ordering a pizza with
3 optional toppings {cheese, corn, olives}. You can pick NO topping ∅, pick 1, pick 2, or take all 3! Every possible combination you can build is a Subset — and the mega-menu containing ALL 2³ = 8 possible orders is the Power Set P(A)!SUBSET (A ⊂ B)Every item in
A is also inside B. ∅ is a subset of EVERY set!POWER SET P(A)Set of ALL subsets. If
n(A) = m, then n[P(A)] = 2ᵐ!REAL INTERVALS
(a, b) excludes endpoints (open <); [a, b] includes endpoints (closed ≤)!🧠 Memory Trick: Think of [ ] square brackets like sharp square claws that GRAB the boundary points; ( ) round brackets are slippery and let boundaries slide away!
💡 Simple Kid-Check: For each of the
m items in set A, you make a simple YES/NO choice ("put it in the subset or leave it out?"). With m items, you have 2 × 2 × ... × 2 = 2ᵐ total subsets!| Interval Notation | Set-Builder Meaning | Endpoints Included? | Length (b − a) |
|---|---|---|---|
(a, b) Open | {x ∈ ℝ : a < x < b} | Neither a nor b | b − a |
[a, b] Closed | {x ∈ ℝ : a ≤ x ≤ b} | Both a and b | b − a |
[a, b) Half-open | {x ∈ ℝ : a ≤ x < b} | a only (not b) | b − a |
(a, b] Half-open | {x ∈ ℝ : a < x ≤ b} | b only (not a) | b − a |
Watch
∈ vs ⊂! If A = {1, 2}, then 1 ∈ A (element!), whereas {1} ⊂ A (subset with braces!). And remember how many subsets does ∅ have? 2⁰ = 1, i.e., P(∅) = {∅}! Wait: how many elements in P(P(∅))? 2¹ = 2!Topic 1.7 – 1.8: Universal Set & Interactive Venn Diagrams
Draw your whole school as a big rectangle
U (the Universal Set). Inside that playground rectangle, draw one circle for students playing Football (A) and another overlapping circle for Cricket (B). Immediately your brain SEES who plays both sports, only one, or neither — that superhero visual map is a Venn Diagram!UNIVERSAL SET (U)Outer rectangle holding every element under discussion for that problem
SUBSETSCircles inside
U. Nested circle = subset A ⊂ B; non-touching circles = disjoint!WHO NAMED IT?English logician John Venn (1834–1923) — visual proofs for Set Algebra!
🧠 Memory Trick: Rectangle = Universe boundary; Circles = sets inside it; Overlap lens = members shared by both clubs!
💡 Simple Kid-Check: A Venn diagram turns invisible set formulas into glowing geometry! Just look at the shaded region to know whether an equation is true — no guesswork needed!
- 1Overlapping Circles: Share common items in the centre football-shaped lens (
A ∩ B ≠ ∅). - 2Nested Circles: Circle
Acompletely inside circleBmeansA ⊂ B(e.g. All squares inside rectangles). - 3Disjoint Circles: Separate circles with zero overlap (
A ∩ B = ∅), e.g. Even integers and Odd integers.
Always draw the enclosing rectangle labelled
U around your Venn circles in board exams! Without the outer rectangle U, complement diagrams like A' cannot show the outside region properly and lose marking key credit.Topic 1.9: Operations on Sets: Union, Intersection & Difference
Suppose you have a party guest list
A of school friends and list B of tuition friends. Inviting everyone from EITHER list (counting common friends once!) is Union (A ∪ B). Finding the common friends in BOTH lists is Intersection (A ∩ B). Inviting ONLY school friends who aren't in tuition is Difference (A − B)!UNION (A ∪ B)Items in A OR B (or both): combine & remove duplicates
INTERSECTION (A ∩ B)Items in BOTH A AND B (only the common overlap!)
DIFFERENCE (A − B)In A ONLY, NOT in B! Note:
A − B ≠ B − A in general!🧠 Memory Trick: ∪ looks like a big cup that catches EVERYTHING (OR); ∩ looks like a tunnel/bridge trapping ONLY what crosses BOTH (AND)!
💡 Simple Kid-Check: Think of Set Operations like adding, subtracting, and finding common items with checklists: OR = Union (∪), AND = Intersection (∩), ONLY = Difference (−)!
🎲 Live Calculation with
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A = {1, 2, 3, 4}, B = {3, 4, 5, 6}:•
A ∪ B = {1, 2, 3, 4, 5, 6} (everything combined)•
A ∩ B = {3, 4} (only the shared twins)•
A − B = {1, 2} (chop off anything belonging to B!)•
B − A = {5, 6} (completely different from A − B!)
Notice that
A − B, A ∩ B, and B − A are three mutually disjoint slices! Adding their sizes always gives the full Union size: n(A ∪ B) = n(A − B) + n(A ∩ B) + n(B − A)!Topic 1.10: Complement of a Set, De Morgan's Laws & Counting Formula
If your school has 50 students (
U) and 32 joined the Science fair (A), how many stayed back in class? 50 − 32 = 18! Everyone inside the Universe U who is OUTSIDE A forms the Complement (A' = U − A). And when a complement bar breaks across brackets, De Morgan's Law flips ∪ into ∩ like magic!COMPLEMENT (A')
A' = U − A (everything outside A). Double flip cancels: (A')' = A!DE MORGAN'S LAWS
(A ∪ B)' = A' ∩ B'(A ∩ B)' = A' ∪ B' (Sign FLIPS!)SURVEY COUNT RULE
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)🧠 Memory Trick: De Morgan's Prime Bar acts like an inverter switch — when
' splits onto A and B, it FLIPS the sign between them (∪ ↔ ∩)!💡 Simple Kid-Check: Imagine counting classmates who like Tea (20) OR Coffee (15). If 5 kids love BOTH, simple addition
20 + 15 = 35 double-counts those 5 kids! So subtract the overlap once: 20 + 15 − 5 = 30!
Important Laws:
3-Set Extension:
A ∪ A' = U | A ∩ A' = ∅ | U' = ∅ | ∅' = U3-Set Extension:
n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(C∩A) + n(A∩B∩C)
In "neither Tea nor Coffee" word problems, students forget to subtract from
n(U)! Use De Morgan directly: Neither A nor B = n(A' ∩ B') = n((A ∪ B)') = n(U) − n(A ∪ B)!Solved Previous Year Questions (Kerala DHSE & NCERT)
Q1 (Kerala HSE March • 1 Marks): Write the set A = {2, 4, 6, 8, 10} in Set-Builder form.
Key (1 Mark):
A = {x : x = 2n, n ∈ ℕ and 1 ≤ n ≤ 5} (or {x : x is an even natural number ≤ 10}).Q2 (Kerala HSE Say • 1 Marks): Express the interval [−3, 5) in Set-Builder form.
Key (1 Mark):
{x ∈ ℝ : −3 ≤ x < 5}.Q3 (Kerala HSE March • 2 Marks): Write down all subsets of A = {1, 2, 3}. How many elements are in P(A)?
Subsets (8 total = 2³):
Therefore
∅, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}.Therefore
n[P(A)] = 8.Q4 (NCERT / DHSE Model • 2 Marks): Which of these sets are empty? (a) A = {x : x is an even prime > 2} (b) B = {x ∈ ℝ : x² = 2}
• (a) Empty Set (∅): 2 is the only even prime.
• (b) Not empty:
• (b) Not empty:
B = {−√2, √2}.Q5 (Kerala HSE March • 3 Marks): If U = {1,2,3,4,5,6}, A = {2,3}, B = {3,4,5}, verify that (A ∪ B)' = A' ∩ B'.
A ∪ B = {2,3,4,5} ⇒ LHS = (A ∪ B)' = {1,6}.A' = {1,4,5,6}, B' = {1,2,6}.RHS = A' ∩ B' = {1,6} = LHS. Verified!Q6 (Kerala HSE Say • 2 Marks): Given A = {a, e, i, o, u} and B = {a, i, k, u}, find A − B and B − A.
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A − B = {e, o} (in A only)•
B − A = {k} (in B only)Q7 (Kerala HSE March • 3 Marks): In a group of 400 people, 250 speak Hindi and 200 speak English. How many people speak BOTH Hindi and English if everyone speaks at least one?
n(H ∪ E) = n(H) + n(E) − n(H ∩ E)400 = 250 + 200 − n(H ∩ E) ⇒ n(H ∩ E) = 450 − 400 = 50 people.Q8 (Kerala HSE Model • 3 Marks): In a survey of 600 students, 150 drink tea, 225 drink coffee, and 100 drink both. Find how many drink NEITHER tea nor coffee.
n(T ∪ C) = 150 + 225 − 100 = 275.Neither tea nor coffee =
n(U) − n(T ∪ C) = 600 − 275 = 325 students.