Set Theory is one of the most scoring chapters in CDS & OTA Maths: the rules are few, the diagrams are visual, and most questions reduce to a single counting formula. At The Cavalier we drill candidates on Venn diagrams and the inclusion−exclusion principle because a well-drawn diagram turns a confusing word problem into a 30-second answer. This guide builds the topic from first principles.
Why Set Theory is a guaranteed scorer
A set is simply a well-defined collection of distinct objects, called its elements or members. "Well-defined" means there is a clear rule to decide whether any object belongs to the set or not. The collection of vowels in English is a set; the collection of "tall students" is not, because "tall" is vague.
In the CDS Elementary Mathematics paper you can expect 1–3 questions from sets in almost every sitting. They are popular with examiners because the same small toolkit — notation, Venn diagrams and one counting formula — solves nearly every variant. Unlike trigonometry or geometry, there is very little to memorise here, so the return on study time is exceptionally high. A candidate who is comfortable drawing a clean Venn diagram rarely loses a mark on this chapter.
Set Theory also forms the foundation for relations, functions and probability, which appear later in the syllabus. So time spent mastering the basics here pays off across several topics, not just one. The concepts trace back to the work of the mathematician Georg Cantor, but for CDS purposes you only need the practical counting and diagram skills covered below.
Sets are usually named with capital letters (A, B, C) and elements with small letters. We write a ∈ A to mean "a is an element of A" and a ∉ A for "a is not an element of A". Standard number sets have fixed symbols: N for naturals, Z for integers, Q for rationals and R for real numbers.
Two ways to write a set
There are two standard methods of describing a set.
Roster (tabular) form
List every element inside curly braces, separated by commas: A = {2, 3, 5, 7}. Order does not matter and elements are never repeated, so {1, 2, 2, 3} is simply written {1, 2, 3}.
Set-builder form
State the common property the elements share: A = {x : x is a prime number less than 10}. Read the colon as "such that". This form is essential when a set has infinitely many members, for example N = {x : x is a natural number}.
If a CDS question gives a set in set-builder form, your first move should usually be to rewrite it in roster form. Listing the actual elements removes ambiguity and prevents silly mistakes.
Types of sets you must know
CDS examiners love testing the precise definitions below.
- Empty (null) set: a set with no elements, written ∅ or { }. Note { } and {0} are different — {0} contains one element, the number zero.
- Singleton set: a set with exactly one element, e.g. {5}.
- Finite set: the counting of elements ends, e.g. days of the week.
- Infinite set: the counting never ends, e.g. the set of integers.
- Equal sets: A = B if they have exactly the same elements.
- Equivalent sets: same number of elements (same cardinality) but not necessarily the same elements.
- Universal set (U): the master set containing all elements under discussion.
The empty set is not nothing — it is a valid set. Its cardinality n(∅) = 0, but ∅ itself is a member of every power set.
Subsets, supersets and power sets
A is a subset of B (written A ⊆ B) if every element of A is also in B. If A ⊆ B but A ≠ B, then A is a proper subset, written A ⊂ B. Here B is called a superset of A.
A set with n elements has exactly 2n subsets and 2n − 1 proper subsets. The collection of all subsets is the power set P(A), and n(P(A)) = 2n.
Two facts trip students up: the empty set ∅ is a subset of every set, and every set is a subset of itself. So a 3-element set {a, b, c} has 23 = 8 subsets, including ∅ and {a, b, c} itself. Listed out, these are ∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c} and {a, b, c}.
A useful related count is the number of subsets of a given size. The number of r-element subsets of an n-element set equals the combination nCr. For example, a 4-element set has 4C2 = 6 two-element subsets. Adding the counts for every size from 0 to n always returns the grand total 2n, which is a neat way to cross-check your work.
The core set operations
Four operations generate almost every CDS question.
- Union (A ∪ B): all elements in A, or B, or both.
- Intersection (A ∩ B): elements common to both A and B.
- Difference (A − B): elements in A but not in B.
- Complement (A′ or Ac): elements of the universal set U that are not in A, so A′ = U − A.
Two sets with no common element (A ∩ B = ∅) are called disjoint sets. The symmetric difference A Δ B collects elements in exactly one of the two sets, that is (A − B) ∪ (B − A); it is the union minus the intersection.
These operations obey simple algebra-like laws. Union and intersection are both commutative (A ∪ B = B ∪ A) and associative, and each distributes over the other: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). You rarely need to quote these by name in CDS, but recognising them helps you simplify expressions quickly.
De Morgan's laws are favourites: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′. The complement "flips" union into intersection.
Reading Venn diagrams
A Venn diagram shows the universal set as a rectangle and each set as a circle inside it. Overlapping circles show common elements. Most CDS survey problems involve two or three overlapping circles.
For two sets, three regions exist: only A, only B, and the overlap A ∩ B. For three sets there are seven regions, including the central A ∩ B ∩ C where all three overlap. Filling these regions one by one — starting from the centre and working outward — is the fastest reliable method.
Here is the working order for a typical three-set survey. Suppose 5 people use all three products. Put 5 in the central region. If 12 use products A and B, then the "A and B but not C" region is 12 − 5 = 7. Repeat for the other two pairwise overlaps. Finally, the "only A" region equals n(A) minus everything else already placed inside circle A. Working from the centre outward like this guarantees no element is counted twice.
Always sketch the Venn diagram even on rough paper. Place the innermost (all-three) value first, then subtract to find each pairwise-only and single-set region. This avoids double counting.
The counting (inclusion-exclusion) formulas
The number of elements in a set A is its cardinality, written n(A). The counting formulas below are the heart of this chapter.
Two sets:
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Three sets:
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C)
The logic is simple: when we add n(A) + n(B), the overlap gets counted twice, so we subtract it once. For three sets we add back the triple overlap because it was over-subtracted. "Neither/none" problems use n(U) − n(A ∪ B).
Worked example: two-set survey
In a class of 60 students, 30 play cricket, 25 play football and 12 play both. How many students play (i) at least one game, and (ii) neither game?
So 43 students play at least one game and 17 play neither. Notice how a single formula handles both parts of the question.
Worked example: three-set problem
In a survey of 100 people: 40 read newspaper A, 35 read B, 30 read C; 12 read A and B, 9 read B and C, 8 read A and C, and 5 read all three. How many read at least one newspaper?
So 81 people read at least one newspaper, and 100 − 81 = 19 read none. Always add the single-set totals first, subtract the three pairwise overlaps, then add back the triple overlap. If the question further asks how many read exactly one newspaper or exactly two, build the full Venn diagram and read those regions off directly rather than hunting for a separate formula.
Mistakes that cost easy marks
Forgetting to add back n(A ∩ B ∩ C) in the three-set formula. The sign pattern is plus singles, minus pairs, plus triple. Memorise it.
- Confusing "only A" with "A". The value n(A) includes the overlaps; "only A" excludes them.
- Reading "both" as "either". "Both" means intersection; "either/or" means union.
- Treating ∅ as having one element — it has zero.
- Counting {a, b} as having more subsets than 22 = 4. The four are ∅, {a}, {b}, {a, b}.
Previous-year style question
Q In a group of 200 students, 120 like tea and 90 like coffee. If every student likes at least one of the two drinks, how many students like both tea and coffee?
Answer: Since everyone likes at least one drink, n(T ∪ C) = 200. Using n(T ∪ C) = n(T) + n(C) − n(T ∩ C): 200 = 120 + 90 − n(T ∩ C), so n(T ∩ C) = 210 − 200 = 10 students like both.
This is the classic "every member likes at least one" twist — the key is recognising that n(union) equals the total group size. A common alternative phrasing gives the number who like "only tea" or "only coffee" and asks for the overlap; in that case set up the diagram with three unknown-free regions and solve directly. Whenever a question says "each", "every" or "all students", suspect that the universal set equals the union, which removes one unknown and makes the formula solvable in a single step.
Quick revision
- A set is a well-defined collection of distinct elements; describe it in roster or set-builder form.
- n elements → 2n subsets; ∅ is a subset of every set.
- Operations: union ∪, intersection ∩, difference −, complement A′.
- Two sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
- Three sets: add singles, subtract three pairs, add back the triple overlap.
- "Neither" = n(U) − n(union). Always sketch a Venn diagram.
Practise 20–30 PYQs and the survey-style questions will become almost mechanical. This is one chapter where The Cavalier expects every aspirant to score full marks.
Frequently asked questions
How many questions on sets appear in the CDS Maths paper?
Typically one to three questions per paper, most often a survey-type word problem solved by the inclusion-exclusion formula, plus an occasional theory question on subsets or operations.
What is the difference between equal and equivalent sets?
Equal sets have exactly the same elements, while equivalent sets only need the same number of elements (same cardinality). All equal sets are equivalent, but not all equivalent sets are equal.
Why do we add back the triple intersection in the three-set formula?
Elements common to all three sets get subtracted once too often when we remove the three pairwise overlaps, so we add n(A ∩ B ∩ C) back once to count them correctly.
Is the empty set a subset of every set?
Yes. The empty set ∅ is a subset of every set, including itself, because it contains no element that could violate the subset condition. It is also a member of every power set.
How many subsets does a set with 5 elements have?
A set with 5 elements has 2 to the power 5 = 32 subsets in total, of which 31 are proper subsets (all except the set itself).
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