Sets, Relations and Functions is the very first chapter of NDA Maths, and it quietly powers everything that follows — from trigonometry to calculus. Once you can read symbols like ∪, ∩ and ⊆ fluently, these questions become almost free marks. In this Cavalier guide you will learn every definition, formula and trick the exam actually tests.
Why This Topic Matters for NDA
Almost every NDA paper carries 3 to 6 direct questions from sets, relations and functions. These are short, formula-based questions where a clear head beats heavy calculation. Even better, the language of this chapter — symbols, domain, range, mapping — reappears in coordinate geometry, calculus and probability.
Master it early and two things happen: you bank easy marks, and the rest of the syllabus suddenly reads more easily because you understand its vocabulary. When a calculus question says “find the domain of the function”, it is really asking a sets question in disguise. When probability talks about “favourable outcomes”, it is counting elements of a set. This chapter is the grammar of the whole paper.
The good news for a busy Class 11 or 12 student is that there is very little to calculate here. The marks come from clear definitions, a handful of formulas, and the habit of reading symbols carefully. That makes it the highest return-on-effort topic in the entire NDA Maths syllabus.
NDA does not give you a formula sheet. You must memorise the cardinality formulas and the conditions for each type of relation and function.
What Exactly Is a Set?
A set is a well-defined collection of distinct objects. “Well-defined” means there is no doubt whether an object belongs or not. The objects inside are called elements or members. The word “well-defined” is doing a lot of work here. “The collection of all tall boys in your class” is NOT a set, because “tall” is a matter of opinion. But “the collection of all boys taller than 170 cm” is a set, because there is a clear yes/no test for membership.
We write sets in two ways:
- Roster form: list the elements, e.g. A = {2, 4, 6, 8}.
- Set-builder form: describe a rule, e.g. A = {x : x is an even number, x ≤ 8}.
If an element a belongs to set A we write a ∈ A; if it does not, a ∉ A. The number of distinct elements in a finite set A is its cardinal number, written n(A).
Repetition and order never matter in a set. {1, 2, 2, 3} is the same as {1, 2, 3}, and n of that set is 3, not 4.
Important Types of Sets
NDA loves to test whether you can name a set correctly. Learn these standard types:
- Empty (null) set: has no element, written ∅ or { }. Note: {0} and {∅} are NOT empty.
- Singleton set: has exactly one element, e.g. {5}.
- Finite set: countable number of elements.
- Infinite set: unending, e.g. the set of natural numbers N.
- Equal sets: A = B if they have exactly the same elements.
- Equivalent sets: A and B with the same number of elements, n(A) = n(B), even if elements differ.
- Universal set (U): the big set containing all elements under discussion.
The empty set ∅ is a subset of every set, but it is an element of a set only if it is actually listed inside, like in {∅, 1}.
Subsets, Power Set and Intervals
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 (A ⊂ B).
If a set has n elements, then:
- Number of subsets = 2n
- Number of proper subsets = 2n − 1
- Number of elements in the power set P(A) = 2n
The power set P(A) is the set of all subsets of A, including ∅ and A itself. For example, if A = {1, 2}, then P(A) = {∅, {1}, {2}, {1, 2}}, which has 22 = 4 members — exactly as the formula predicts.
On the number line, sets of real numbers are often written as intervals: [a, b] includes the endpoints (closed), while (a, b) excludes them (open). A half-open interval like [a, b) includes a but not b. You will meet these constantly when finding the domain and range of functions later in this chapter, so get comfortable with the bracket notation now.
Operations on Sets
Four operations carry most of the marks. Picture each one as a Venn diagram.
- 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′): elements of the universal set U that are not in A, so A′ = U − A.
Two sets with nothing in common (A ∩ B = ∅) are called disjoint sets.
De Morgan’s Laws (very frequently asked):
- (A ∪ B)′ = A′ ∩ B′
- (A ∩ B)′ = A′ ∪ B′
The Counting (Cardinality) Formulas
These formulas turn wordy “how many students play both games” problems into one-line solutions.
For two sets:
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
For three sets:
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C)
If the sets are disjoint, the intersection terms vanish and n(A ∪ B) simply equals n(A) + n(B). Always start three-set problems from the centre of the Venn diagram and work outward, filling in the “all three” region first, then the “exactly two” regions, and finally the “only one” regions.
One more handy result the examiner likes: the number of elements belonging to exactly one of two sets is n(A) + n(B) − 2·n(A ∩ B). Spotting whether a question asks for “at least one” or “exactly one” is often the entire difficulty of the problem.
Ordered Pairs and the Cartesian Product
An ordered pair (a, b) keeps order: (2, 3) is different from (3, 2). The Cartesian product A × B is the set of all ordered pairs whose first element comes from A and second from B.
If n(A) = p and n(B) = q, then n(A × B) = p × q. The number of possible relations from A to B is 2pq.
Example: if A = {1, 2} and B = {x, y}, then A × B = {(1, x), (1, y), (2, x), (2, y)}, which has 2 × 2 = 4 pairs.
A × B ≠ B × A unless A = B. The order of the sets matters just like the order inside a pair.
Relations and Their Types
A relation R from A to B is simply any subset of A × B. It pairs up elements that satisfy some rule, like “is less than” or “is a factor of”. When the relation is on a single set A (from A to A), it can have these special properties:
- Reflexive: every element relates to itself, (a, a) ∈ R for all a.
- Symmetric: if (a, b) ∈ R then (b, a) ∈ R.
- Transitive: if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R.
A relation that is reflexive, symmetric AND transitive all at once is called an equivalence relation. “Is equal to” and “is parallel to” are classic equivalence relations.
“Is perpendicular to” is symmetric but NOT reflexive (a line is not perpendicular to itself) and NOT transitive, so it is not an equivalence relation. Check all three conditions every time.
Functions: A Special Kind of Relation
A function f from A to B is a relation in which every element of A is paired with exactly one element of B. No element of A may be left out, and none may point to two different outputs.
- Domain: the set A of all inputs.
- Co-domain: the full set B of possible outputs.
- Range: the set of outputs actually produced — always a subset of the co-domain.
Think of a function as a reliable machine: feed in one value and you always get back one predictable result. The number 3 can never come out as both 9 and 10 from the same machine. This “one input, one output” rule is the single idea that separates functions from ordinary relations, and it is tested again and again.
The vertical line test: if any vertical line cuts a graph more than once, it is NOT a function. Useful when NDA shows graphs.
Types of Functions
Classifying functions is a guaranteed exam favourite:
- One-one (injective): different inputs give different outputs. If f(a) = f(b) forces a = b, it is one-one.
- Onto (surjective): every element of the co-domain is hit, so range = co-domain.
- Bijective: both one-one and onto. Only bijective functions have an inverse.
- Many-one: two or more inputs share the same output.
If n(A) = m and n(B) = n, the number of functions from A to B is nm. The number of one-one functions (when n ≥ m) is n × (n−1) × … down to (n−m+1).
How many functions can be defined from A = {1, 2, 3} to B = {p, q}? How many of them are onto?
A Full Worked Counting Problem
Set problems often arrive as survey questions. Here is the method, step by step.
In a class of 60 students, 35 like cricket, 25 like football, and 10 like both. How many like neither game?
So 10 students like neither. Notice how the union formula did all the heavy lifting in a single line.
Mistakes That Cost Easy Marks
The Cavalier classroom sees the same slips every batch. Avoid these:
- Confusing ⊆ (subset) with ∈ (element). 2 ∈ {2, 3} is correct, but {2} ⊆ {2, 3}.
- Forgetting that the empty set is a subset of every set when counting subsets.
- Assuming A × B = B × A — it does not unless the sets are equal.
- Calling a relation an equivalence relation after checking only one or two properties.
- Mixing up co-domain and range when deciding if a function is onto.
n(A ∪ B) is NOT just n(A) + n(B). You must subtract the overlap n(A ∩ B), otherwise common elements get counted twice.
Previous-Year Style Question
Q. If a set A has 4 elements, then the number of proper subsets of A is:
Answer: Number of subsets = 24 = 16. Proper subsets exclude the set itself, so 16 − 1 = 15. (If a question instead asks for subsets excluding both A and the empty set, the answer would be 14.)
Read the wording carefully: “proper subsets”, “non-empty subsets” and “subsets” all give slightly different counts. Examiners deliberately test this.
Quick Revision
- Set = well-defined collection of distinct objects; n(A) is its cardinal number.
- Subsets of an n-element set = 2n; power set P(A) also has 2n members.
- n(A ∪ B) = n(A) + n(B) − n(A ∩ B); learn the three-set version too.
- De Morgan: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.
- Equivalence relation = reflexive + symmetric + transitive, all three.
- Function pairs every input with exactly one output; bijective = one-one + onto.
- Functions from A(m) to B(n) = nm; relations from A to B = 2pq.
Drill five PYQs from this chapter every week and these marks will become automatic in the real NDA paper.
Frequently asked questions
How many questions come from Sets, Relations and Functions in NDA Maths?
Typically 3 to 6 direct questions appear per paper, often as quick formula-based or definition-based questions. They are among the most scoring questions if your basics are clear.
What is the difference between a relation and a function?
Every function is a relation, but not every relation is a function. A relation becomes a function only when each element of the domain is mapped to exactly one element of the co-domain, with no input left unmapped.
Is the empty set a subset of every set?
Yes. The empty set is a subset of every set, including itself. That is why a set with n elements has 2 to the power n subsets, counting the empty set and the full set.
What makes a relation an equivalence relation?
A relation must be reflexive, symmetric and transitive all at the same time to be an equivalence relation. Examples include is equal to and is parallel to.
When does a function have an inverse?
A function has an inverse only when it is bijective, meaning it is both one-one (injective) and onto (surjective). If even one of these fails, the inverse does not exist.
Which books does The Cavalier recommend for this topic?
Start with NCERT Class 11 and Class 12 Mathematics for theory, then practise from a topic-wise NDA Maths PYQ collection on Sets, Relations, Functions and Binary Numbers to master the exam pattern.
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