Sets
1. What This Chapter Covers
The chapter opens with a question that is not about mathematics at all: when you are asked to describe a person, how would you do it?
The book's four examples answer it the same way each time:
- Ramanujan was a mathematician, interested in number theory.
- Dasarathi was a Telugu poet and also a freedom fighter.
- Albert Einstein was a physicist who proposed the theory of relativity.
- Maryam Mirzakhani is the only woman mathematician to win the Fields Medal.
Every one of those puts the person into a larger recognisable group first — mathematician, poet, physicist — and only then adds the specific detail. That is what classification is, and the book points out we do it everywhere: library books are shelved by subject, chemical elements are sorted into groups and classes, and your own mathematics syllabus is cut into 14 chapters under different headings.
The book's fourth example is no longer accurate. Maryna Viazovska won the Fields Medal in 2022, two years before this impression was printed, so Maryam Mirzakhani is the first woman to win it, not the only one. The name is also printed "Mirzakhan". Reproduced here as printed, with the correction noted.
The opening box gives a real classification to hold on to: the dental formula. Human permanent teeth divide into four types by chewing function — incisors, canines, premolars and molars — and the formula for that set is 2, 1, 2, 3. The teeth come back three more times in this chapter, as the example for universal sets and subsets.
The chapter is allotted 8 periods in June and runs from textbook page 28 to page 50.
2. What a Set Is
Before any notation, the book lists the number collections you already use:
| Symbol | Collection |
|---|---|
| ℕ | natural numbers 1, 2, 3, … |
| 𝕎 | whole numbers 0, 1, 2, 3, … |
| 𝕀 or ℤ | integers 0, ±1, ±2, ±3, … |
| ℚ | rational numbers, those that can be written as p/q with p, q integers and q ≠ 0 |
| ℝ | real numbers, those that have a decimal expansion |
Then the definition, which is three conditions packed into one line:
A set is a well-defined collection of distinct objects. The objects in a set are called elements. Sets are written by enclosing all of its elements between the brackets { }.
Well-defined is the load-bearing word. It means that for any object you name, there is no argument about whether it is in the collection. "The first five prime numbers" is well-defined: {2, 3, 5, 7, 11}. "The ten most talented writers of India" is not, because two people will disagree about the list — which is exactly why Exercise 2.1 asks you to sort five collections into sets and non-sets.
Distinct is the second condition, and it is the one that shows up in marks. An element is never repeated: the set of letters in the word SCHOOL is {s, c, h, o, l}, not {s, c, h, o, o, l}.
Membership has its own symbol. "Second molar is in the set of molars" is written second molar ∈ M, read "second molar belongs to set M". The negation is square ∉ M, read "square does not belong to the set M".
3. Roster Form and Set-Builder Form
Sets are named with capital letters — A, B, C — and written down in one of two ways.
Roster form lists the elements. M = {first molar, second molar, third molar}. Q = {square, rectangle, rhombus, parallelogram, kite, isosceles trapezium}, the set of quadrilaterals with at least two equal sides.
The book attaches two notes to roster form, and both are examinable:
- The order is immaterial. The digits of the Ramanujan number can be written {7, 2, 1, 9} or {1, 2, 7, 9} or {1, 7, 2, 9}; they are the same set.
- An element is not repeated, as with SCHOOL above.
Roster form breaks down when the set is infinite. Can you write ℚ by listing its elements? The book's Think and discuss box asks exactly that, and the answer is no — which forces the second notation.
Set-builder form describes the elements by a common property instead of naming them.
The colon is always read "such that"; everything after it is the test an object has to pass to get in.
The same syntax writes the rationals compactly: ℚ = {x : x = p/q, p and q are integers and q ≠ 0}.
The book puts four sets side by side in both forms:
| Roster form | Set-builder form |
|---|---|
| V = {a, e, i, o, u} | V = {x : x is a vowel in the English alphabet} |
| A = {−2, −1, 0, 1, 2} | A = {x : −2 ≤ x ≤ 2, x ∈ ℤ} |
| B = {1, 1/2, 1/3, 1/4, 1/5} | B = {x : x = 1/n, n ∈ ℕ, n ≤ 5} |
| C = {2, 5, 10, 17} | C = {x : x = n² + 1, n ∈ ℕ, n ≤ 4} |
Read the last row backwards to check it: n = 1, 2, 3, 4 gives 2, 5, 10, 17.
4. The Empty Set
Some descriptions pass the well-defined test and still catch nothing.
- A = {x : x is a natural number smaller than 1} — there is no such natural number.
- D = {x : x is an odd number divisible by 2} — there is no such number either.
A set which does not contain any element is called an empty set, or a Null set, or a void set. It is denoted by φ or { }.
Three more of the book's examples, each empty for a different reason: {x : 1 < x < 2, x is a natural number}; {x : x² − 2 = 0 and x is rational}, which is empty because √2 is irrational — the result proved in chapter 1; and {x : x² = 4, x is an odd number}.
The book then prints a Note that is worth its own figure, because it is the single most common slip in the chapter.
Zero is a perfectly good element; a box containing a zero is not an empty box.
5. The Universal Set and Venn Diagrams
Go back to the teeth. You split the whole set into incisors, canines, premolars and molars — but are the molars still members of the whole teeth set? Obviously yes. The whole teeth set is the universal set of those four.
The universal set is the frame of reference: everything under discussion in a particular problem lives inside it. If you are studying groups of people in Telangana, the universal set is all the people in Telangana; if you are studying groups in India, it is everyone in India.
The universal set is generally denoted by μ, and sometimes by U. It is usually drawn as a rectangle.
Using μ for the universal set is this book's convention and is worth noticing, because other books write U or ξ for the same thing.
A Venn diagram — properly a Venn-Euler diagram — draws the universal set as a rectangle and the sets inside it as closed curves, usually circles.
The book asks what the empty part of the diagram represents; it is every member of μ that is not in A.
Two symbols get introduced in a side box here and are used from now on. "If x < 3 then x < 4" is written with a one-way implication arrow, and "x − 2 = 5 exactly when x = 7" with a two-way implication, read "if and only if" and often shortened to iff.
6. Subsets
Take A = {1, 2, 3}. How many sets can you build using as many of its elements as you like?
There are eight: { }, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3} and {1, 2, 3}. Every one of them is a subset of A — including the empty set at one end and A itself at the other.
If all elements of set A are present in B, then A is said to be a subset of B, written A ⊆ B. Equivalently, A ⊆ B if and only if a ∈ A implies a ∈ B.
Two consequences are boxed in the book:
Null set is a subset of every set. If φ were not a subset of A, it would have to contain an element that is not in A — and it contains no elements at all.
Every set is a subset of itself. Every element of A is an element of A.
When B sits inside A but does not fill it, B is a proper subset. The vowels V = {a, e, i, o, u} are a proper subset of the alphabet A = {a, b, c, …, z}, because every vowel is a letter but not every letter is a vowel.
The book states on page 37 that ⊂ denotes a proper subset and ⊆ denotes a subset. It then uses ⊂ loosely for "subset" elsewhere — writing A ⊂ A for "every set is a subset of itself" on page 36, and defining subset with ⊂ in the chapter summary. Strictly, a set is never a proper subset of itself. Read ⊂ in this book as "subset" unless the sentence is drawing the distinction.
Sharing some elements is not enough; a subset needs every one of its elements to be in the larger set.
The number systems give a chain of subsets: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ, and also ℚ' ⊂ ℝ, where ℚ' is the set of irrationals — every real that is not rational, so ℚ' = {x : x ∈ ℝ and x ∉ ℚ}. But ℕ ⊄ ℚ': no natural number is irrational.
7. Union, Intersection and Difference
Arithmetic has addition, subtraction, multiplication and division. Sets have three operations of their own, and the book introduces all three through one situation: μ is everyone in your school, A is the students in your class absent on Tuesday, B those absent on Wednesday.
A = {Roja, Ramu, Ravi} and B = {Ramu, Preethi, Haneef}.
Union. Who was absent on Tuesday or Wednesday? Roja, Ramu, Ravi, Haneef and Preethi — but not Akhila, who is always present.
A ∪ B = {x : x ∈ A or x ∈ B}, read "A union B".
Intersection. Who was absent on Tuesday and Wednesday? Only Ramu.
A ∩ B = {x : x ∈ A and x ∈ B}, read "A intersection B".
Difference. Which odd numbers under 10 are not prime? With A = {1, 3, 5, 7, 9} and B = {3, 5, 7}, the answer is {1, 9}.
A − B = {x : x ∈ A and x ∉ B}, read "A difference B" or "A minus B".
Union takes everything either circle holds, intersection only the overlap, difference only the part of A the overlap leaves behind.
Four results follow from the definitions and are worth holding:
- The common element is written once. {2, 5, 6, 8} ∪ {5, 7, 9, 1} = {1, 2, 5, 6, 7, 8, 9}, not eight elements.
- If B ⊂ A then A ∪ B = A. Adding a subset back in adds nothing new.
- Sets with no common element are disjoint, and for them A ∩ B = φ. {1, 3, 5, 7} and {2, 4, 6, 8} are disjoint; their Venn circles are drawn apart.
- A − B ≠ B − A. With A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7}, A − B = {1, 2, 3} while B − A = {6, 7}. Subtraction of sets is no more symmetric than subtraction of numbers.
A Think and discuss box adds a fourth observation to check for yourself: A − B, B − A and A ∩ B are mutually disjoint, and together they make up A ∪ B.
8. Equal Sets
Three sets of cricketers:
- A = {Sachin, Dravid, Kohli}
- B = {Dravid, Sachin, Dhoni}
- C = {Kohli, Dravid, Sachin}
A and C contain exactly the same players in a different order, so they are equal. A and B are not, because Kohli is in one and Dhoni in the other.
Two sets A and C are equal if every element of A belongs to C and every element of C belongs to A — that is, A ⊆ C and C ⊆ A. Written A = C.
The two-way condition is the whole definition, and it is why equality is usually proved in two halves. Example 11 is the model: C is the set of letters in ASSASSINATION and B the set of letters in STATION. Written in roster form with repeats dropped, both are {A, S, I, N, T, O}, so C ⊆ B and B ⊆ C, hence C = B.
Example 9 is the contrast: A = {1, 2, 3} and B = {1, 2, 3, 4}. Here A ⊂ B but B ⊄ A, so A ≠ B. One-way containment is not equality.
Example 10 uses the same test on two differently-described sets: the primes smaller than 6 are {2, 3, 5}, and the prime factors of 30 are also 2, 3 and 5, so the two sets are equal despite having nothing in common in their wording.
9. Finite Sets, Infinite Sets and Cardinality
Some sets you can count to the end of, and some you cannot.
- A = {the students of your school} — countable, however large the school.
- L = {2, 3, 5, 7} — four elements.
- B = {x : x is an even number} — no end.
- J = {x : x is a multiple of 7} — no end either.
A set whose number of elements can be expressed as a definite whole number is a finite set. Otherwise it is an infinite set.
The book's further examples separate the idea from mere size: the days of the week form a finite set; the solutions of x² − 16 = 0 form a finite set with two elements; but the points on a line, or the lines through a single point, form infinite sets.
Example 13 is a useful drill because the answers do not match first impressions. {x : x ∈ ℕ and x² = 4} looks like it should have two elements, but −2 is not a natural number, so the set is {2} and finite. {x : x ∈ ℕ and 2x − 2 = 0} is {1}, finite. {x : x ∈ ℕ and x is prime} is infinite, because the primes never run out.
For a finite set, the count itself gets a name.
The number of elements in a finite set is called the cardinal number, or the cardinality, of the set, written n(A).
With A = {1, 2, 4}, B = {6, 7, 8, 9, 10} and C = {x : x is a letter in the word INDIA}, we get n(A) = 3, n(B) = 5 and n(C) = 4 — because I repeats in INDIA and elements of a set are distinct, so C = {I, N, D, A}.
The note that follows is the counterpart of the φ versus {0} warning: n(φ) = 0. The empty set is finite, and its count is zero.
Example 14 then sets a trap deliberately. With A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}, n(A) = 5 and n(B) = 4, yet A ∪ B = {1, 2, 3, 4, 5, 6, 8} has seven elements, not nine.
The two elements in the overlap belong to both sets, so adding the counts charges for them twice.
The book leaves the fix as its closing Think and discuss: what is the relation between n(A), n(B), n(A ∩ B) and n(A ∪ B), and what happens when A and B are disjoint?
Working the example backwards answers both. Here n(A ∩ B) = 2, and 5 + 4 − 2 = 7, so n(A ∪ B) = n(A) + n(B) − n(A ∩ B). When the sets are disjoint that last term is zero and the counts simply add.
Note that the book poses this only as a question and never states the formula as a result, so quote it as reasoning from Example 14 rather than as a textbook theorem.
10. What the Exercises Ask, and What the Book Answers
The chapter carries four numbered exercises, and unusually for this book all four have printed answers, at textbook pages 371 to 373.
- Exercise 2.1 (8 questions) is about notation: which collections are sets, filling ∈ or ∉, writing statements in symbols, true-or-false with justification, and converting between roster and set-builder form in both directions.
- Exercise 2.2 (7 questions) drills union, intersection and difference, including the ten-part question on A − B, A − C, A − D and so on for four given sets.
- Exercise 2.3 (5 questions) is equality and subsets, ending with listing all the subsets of five given sets — the answer for a four-element set runs to sixteen.
- Exercise 2.4 (2 questions) sorts sets into empty or not, and finite or infinite.
Two answers in that key are wrong, and both are in Exercise 2.2 question 5, where A is the naturals, B the even naturals, C the odd naturals and D the primes:
- B ∩ D is printed as "{even natural number}". The even naturals and the primes share exactly one member, so B ∩ D = {2}.
- A ∩ D is printed as {2, 3, 5, 7, 11, …, 97}. A is every natural number and D is every prime, so A ∩ D is the whole infinite set of primes; stopping at 97 has no basis in the question.
Check your own work against the key, but not blindly.
11. Summary
A set is a well-defined collection of distinct objects, written inside braces; membership is ∈ and non-membership ∉. Well-defined means no one can argue about whether a given object is in; distinct means nothing is listed twice.
It can be written in roster form, listing the elements, where order does not matter and repeats are dropped; or in set-builder form, {x : x has some property}, with the colon read "such that". Infinite sets force the second form.
The empty set φ has no elements and n(φ) = 0, and it is not the same as {0}. The universal set μ is the frame everything in a problem sits inside, drawn as a rectangle in a Venn diagram.
A is a subset of B when every element of A is in B. The empty set is a subset of every set, and every set is a subset of itself; a proper subset leaves something out. For the number systems, ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ and ℚ' ⊂ ℝ, but ℕ ⊄ ℚ'.
Three operations: A ∪ B collects elements in either set, A ∩ B only those in both, and A − B those in A but not B. Sets with A ∩ B = φ are disjoint, and A − B is generally not B − A.
Two sets are equal when each is a subset of the other — a two-way test, which is why one-way containment such as {1, 2, 3} ⊂ {1, 2, 3, 4} does not make them equal.
A set is finite if its element count is a whole number and infinite otherwise, and for a finite set that count is its cardinality n(A). Because shared elements belong to both sets, n(A ∪ B) is not n(A) + n(B) unless the sets are disjoint.
