Why Relations and Functions Matter So Much 🎯

Think of mathematics as a language that describes patterns. Relations and functions are the grammar rules of that language:

• In coordinate geometry, you describe curves using functions.
• In calculus, you differentiate and integrate functions.
• In probability and statistics, you use functions to model real-life data.
• In computer science, mapping inputs to outputs is exactly what functions do.

So this “Relation and Function Set‑1” is not just a chapter; it is the foundation for almost every higher topic you will study in Class 12 and beyond.

Warm-up: Sets and Cartesian Product 🔄

To talk about relations and functions, we first need sets and their Cartesian product.

1. Quick recap of sets

A set is a well-defined collection of objects.

Examples:

• Set of even natural numbers less than 10: {2, 4, 6, 8}
• Set of letters in the word “MATH”: {M, A, T, H}

2. Cartesian product of sets

If A and B are two sets, then the Cartesian product of A and B, written as A × B, is the set of all ordered pairs (a, b) such that a is in A and b is in B.

For example, let

A = {1, 2} and B = {x, y}

Then

A × B = {(1, x), (1, y), (2, x), (2, y)}

In Class 12 exams, questions often ask you to:

• Find A × B.
• Find its subset which defines a relation.
• Count number of possible relations or functions from A to B.

What Exactly Is a Relation? 🤝

Intuitive idea

A relation between two sets A and B tells us which elements of A are “connected” to which elements of B.

Formally, a relation R from A to B is any subset of A × B.

That means:

• First you form A × B.
• Then you pick some (or all, or none) of the ordered pairs from A × B.
• The set of chosen ordered pairs is your relation.

Example 1: Relation as a subset

Let

A = {1, 2, 3}
B = {4, 5}

Then

A × B = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}

Define a relation R “is less than” from A to B, i.e., a is related to b if a is less than b.

Pairs that satisfy this:

R = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}

Here, since every possible pair satisfies the condition, R is actually equal to A × B.

Types of Relations on a Set 🧩

For CBSE and JEE pattern questions, you must master three key properties of relations defined on a single set A (that is, relations from A to A):

• Reflexive
• Symmetric
• Transitive

When a relation has all three, it is called an equivalence relation.

Assume A is a non-empty set and R is a relation on A.

1. Reflexive relation

R is reflexive if every element of A is related to itself.

In words: for every a in A, (a, a) belongs to R.

Example:

A = {1, 2, 3}

R = {(1, 1), (2, 2), (3, 3), (1, 2)}

This relation is reflexive because all (1, 1), (2, 2), (3, 3) are present.

Non-example:

If R = {(1, 1), (2, 2)} on A = {1, 2, 3}, it is not reflexive because (3, 3) is missing.

2. Symmetric relation

R is symmetric if for every pair (a, b) in R, the pair (b, a) is also in R.

Example:

A = {1, 2, 3}

R = {(1, 2), (2, 1), (2, 3), (3, 2)}

Whenever one pair exists, its reverse also exists, so R is symmetric.

Non-example:

R = {(1, 2), (2, 3)} is not symmetric because (2, 1) and (3, 2) are missing.

3. Transitive relation

R is transitive if whenever (a, b) and (b, c) are in R, then (a, c) must also be in R.

Example:

A = {1, 2, 3}

R = {(1, 2), (2, 3), (1, 3)}

Here (1, 2) and (2, 3) are in R, and (1, 3) is also in R, so it is transitive.

Non-example:

R = {(1, 2), (2, 3)} is not transitive because (1, 3) is not in R.

Quick Revision Box 🧠

Key relation properties on a set A:

• Reflexive: All (a, a) are present.
• Symmetric: Along with (a, b), (b, a) must also be present.
• Transitive: If (a, b) and (b, c) are there, (a, c) must be present.

Equivalence relation: A relation that is reflexive, symmetric, and transitive.

These properties are frequently tested in CBSE Board exams as 2- and 3-mark questions and also appear in conceptual multiple-choice questions in JEE Main.

What Is a Function? 🎛️

Every function is a relation, but not every relation is a function.

A function f from set A to set B is a special type of relation where:

1. Every element of A has an image in B (nothing is left out).
2. No element of A has more than one image (no element is confused between two outputs).

In other words, each input must give you a single, well-defined output.

If f is a function from A to B, we write it as:

Here:

• A is called the domain.
• B is called the codomain.
• The set of all images actually attained is called the range.

Example 2: Checking if a relation is a function

Let A = {1, 2, 3} and B = {4, 5, 6}.

Consider relation R1:

R1 = {(1, 4), (2, 5), (3, 6)}

Check:

• Every element of A appears exactly once in the first position.
• 1 goes to 4, 2 goes to 5, 3 goes to 6.

So R1 is a function.

Now consider relation R2:

R2 = {(1, 4), (1, 5), (2, 6)}

Here:

• Element 1 has two images: 4 and 5.
• Element 3 has no image at all.

So R2 is not a function.

Domain, Codomain and Range Explained Clearly 🌈

Take the function

defined by

Then:

• Domain of f is {1, 2, 3}.
• Codomain of f is {1, 4, 9, 16} (given in the definition).
• Range of f is the set of actual outputs produced.

Compute outputs:

• f(1) = 1
• f(2) = 4
• f(3) = 9

So range of f is {1, 4, 9}. Note that 16 is in the codomain but not in the range.

This difference between codomain and range is a popular short-answer concept in Class 12 exams.

Very Important: One–one, Onto and Bijective Functions 🧷

These ideas appear again in Inverse Trigonometric Functions, Matrices, Determinants and even in higher studies like linear algebra.

1. One–one (Injective) function

A function f from A to B is one–one if different elements of A always go to different elements of B.

In simpler words: no two different inputs have the same output.

Example:

Function f: {1, 2, 3} → {2, 4, 6} defined by

gives:

• f(1) = 2
• f(2) = 4
• f(3) = 6

All outputs are different, so it is one–one.

2. Onto (Surjective) function

A function f from A to B is onto if every element of B is the image of at least one element of A.

In other words, the range is equal to the codomain.

Example:

Let f: {1, 2, 3} → {2, 4, 6} with f(x) = 2x, as before.

Range = {2, 4, 6} and codomain is also {2, 4, 6}, so f is onto.

3. Bijective function

When a function is both one–one and onto, it is called bijective.

Bijective functions are very important because:

• Only bijective functions have inverses which are also functions.
• Many JEE questions ask you to check or prove bijectivity.

Step-by-Step Solved Examples ✍️

Example 3: Identifying relation type

Let A = {1, 2, 3} and relation R on A be defined by

Check whether R is reflexive, symmetric and transitive.

Step 1: Check reflexive

For reflexive: (1, 1), (2, 2), (3, 3) must all be in R.

From R, we see they are all present, so R is reflexive.

Step 2: Check symmetric

For symmetric: whenever (a, b) is in R, (b, a) must also be in R.

Look at the pairs:

• (1, 2) is there and (2, 1) is also there.
• (2, 1) is there and (1, 2) is also there.
• (1, 1), (2, 2), (3, 3) are of the form (a, a), so they automatically satisfy symmetry.

Thus R is symmetric.

Step 3: Check transitive

For transitive: whenever (a, b) and (b, c) are in R, (a, c) must also be in R.

Check combinations:

• (1, 2) and (2, 1) are in R. Then (1, 1) must be in R — which it is.
• (2, 1) and (1, 2) are in R. Then (2, 2) must be in R — which it is.
• Pairs like (1, 1), (2, 2), (3, 3) with themselves do not create any new condition.

No violation is found, so R is transitive.

Conclusion: R is reflexive, symmetric, and transitive. Therefore, R is an equivalence relation.

Example 4: Checking if a relation is a function and finding range

Let A = {–1, 0, 1, 2} and B = {0, 1, 4}.

Relation R is defined by

Check whether R is a function from A to B. If yes, find its range.

Step 1: Check if every element of A has an image

Elements of A: –1, 0, 1, 2.

In R:

• –1 appears once as first element in (–1, 1).
• 0 appears once in (0, 0).
• 1 appears once in (1, 1).
• 2 appears once in (2, 4).

So every element of A has an image in B.

Step 2: Check if any element has more than one image

For each element of A, there is exactly one ordered pair where it appears as first entry. So there is no confusion.

Therefore, R is a function from A to B.

Step 3: Find range

Range is the set of all second elements of the ordered pairs:

From R, second elements are 1, 0, 1, 4.

So range is {0, 1, 4}.

Concept Table: Relations vs Functions 📊

Common Mistakes to Avoid in Exams ⚠️

1. Confusing range and codomain

   Many students simply write the codomain as the range. Remember:

   • Codomain is what is declared.
   • Range is what actually comes out after applying the function rule.

2. Ignoring elements while checking reflexivity

   To test reflexivity, you must check every element a in A, not just a few.

3. Assuming all relations are functions

   If you see a set of ordered pairs, do not immediately call it a function. First verify that every element of the domain appears exactly once as first coordinate.

4. Forgetting to test all three properties for equivalence relations

   You must check reflexive, symmetric and transitive. Even if one fails, it is not an equivalence relation.

5. Writing unordered pairs instead of ordered pairs

   A relation is a set of ordered pairs (a, b), not {a, b}. The order matters.

Exam Strategy: How to Tackle Relation & Function Questions 📝

• For 1-mark CBSE questions, definitions and simple true/false type properties are common. Memorize precise definitions of relation, function, domain, codomain, range, one–one, onto.
• For 2- and 3-mark questions, you will often be asked to:
  • Show that a given relation is or is not reflexive, symmetric, transitive.
  • Check if a given relation is a function, then find its range.
• For JEE-style MCQs, focus on:
  • Counting number of possible relations or functions between finite sets.
  • Identifying one–one, onto, bijective functions.
  • Functions defined by rules like f(x) = x², f(x) = x + 1 etc., with given domains and codomains.

While practising, always:

• Draw a quick arrow diagram in rough work; it makes properties visual.
• Start by listing domain, codomain and range explicitly.
• For proof-type questions, write clear statements: “For every a in A, …” instead of only giving examples.

Mini Self-Check Quiz (Conceptual) ✅

Try answering these mentally:

1. If A has 3 elements and B has 4 elements, how many elements are in A × B?
2. Can a function from A to B have two different arrows starting from the same element of A and ending at two different elements of B?
3. What must be equal for a function to be onto: codomain and range, or domain and codomain?
4. If a relation on a set A is an equivalence relation, how many properties does it satisfy and what are they called?
5. If a function is bijective, can it have an inverse that is also a function?

Try writing down answers and then checking them from the explanations above.

Where to Go Next 🚀

Once you are comfortable with:

• Cartesian product
• Relations as subsets of A × B
• Properties: reflexive, symmetric, transitive
• Functions and their classification (one–one, onto, bijective)
• Domain, codomain and range

you are ready to handle more advanced problems in:

• Inverse of a function
• Composition of functions
• Real-valued functions and graphs (very important for calculus)
• Questions mixing trigonometric and algebraic functions (common in JEE and other entrances)

Ready to Test Yourself? 🧪

To solidify your understanding, attempt a targeted quiz on this chapter and see where you stand.

Practice “Relations And Functions” Questions Now