Class 12 Maths Ch1 Relations & Functions Notes PDF Download

Relation and Function Class 12 Notes discuss mathematical relations, which are simply ways of showing how two things are connected. It’s like matching items from one group to another.

It then explains functions, which are a special kind of relation. Here, every input has just one output, like each person having one birthday.

Based on CBSE class 12 maths syllabus this chapter covers empty, universal, reflexive, symmetric, transitive and equivalence relations, along with one-one, many-one, onto, and bijective functions. Students also learn composition, invertible functions, and binary operations.

Notes & Study Material PDF

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Ultimately, the article underscores the importance of these concepts. Relation and Function Class 12 Notes pdf help us spot patterns, solve problems, and make sense of real-life situations.

Introduction to Relations

Relation talks about connecting two things, like objects or quantities. Imagine you have two sets; a relation is formed when there’s some link between the elements of those sets. In simple words, a function is a type of relation where each input is linked to only one output.

Types of Relations in Class 12 Maths

1. Empty Relation: An empty relation is when no element in set A is related to any element in A. In other words, there are no connections at all. 

For example: If A = {1, 2, 3, 4}, then R = {(a, b) : a − b = 10} is an empty relation, since no such pairs exist. In general, R = ϕ ⊆ A × A.

2. Universal Relation: A universal relation is when every element of a set is connected to every other element in that set. In simple terms, all possible pairs are related.

For example: R = A × A, where Relation R in set A is said to be a universal relation.

3. Reflexive Relation: A reflexive relation means every element in a set is related to itself. For example, if you have numbers in a set, each number must be paired with itself.

For example: Suppose A = {1, 2, 3}

A reflexive relation on A would include pairs like (1,1), (2,2), (3,3) because each element is related to itself.

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

4. Symmetric Relation: A symmetric relation means that if one element is related to another, then the second element is also related to the first.

For example, if (a, b) is in the relation, then (b, a) must also be there.
For a set A = {1, 2}, a symmetric relation could be R = {(1,2), (2,1)}.

5. Transitive Relation: A transitive relation is where if element 'a' is related to 'b' and 'b' is related to 'c', then 'a' must also be related to 'c' for all elements in the set.

For example, if (a, b) and (b, c) are in the relation, then (a, c) should also be there.

For a set A = {1, 2, 3}

A transitive relation could be R = {(1,2), (2,3), (1,3)}.

6. Equivalence Relation: An equivalence relation is a relation that is reflexive, symmetric, and transitive all at once.

For example, on the set of numbers, the relation “= (equal to)” is an equivalence relation because it satisfies all three properties.

Note: Check out other Class 12 Maths Notes in this link.

Functions

A function is a type of relation where each input is linked to only one output. It’s like a rule that makes sure every element from the first set matches with exactly one element from the second set. In simple terms, no input can point to two different outputs.

Types of Functions - One-one, Many-one, Onto, Bijective

Functions can be one-one (injective), many-one, onto (surjective), or bijective. Other functions like identity, constant, modulus, signum, etc., can be found in Class XI.

1. One-one Function or Injective Function: A one-one function (injective function) is a function where each input has a unique output, and no two different inputs share the same output. 

For example, if f(x) = 2x and the domain is {1, 2, 3}, then the outputs are {2, 4, 6}. No two inputs give the same result, so it’s one-to-one.

2. Onto Function or Surjective Function: An onto function (surjective function) is a function where every element in the codomain has at least one pre-image in the domain. 

For example, if f(x) = x² with domain {–2, –1, 0, 1, 2} and codomain {0, 1, 4}, then each element of the codomain is hit by some input. So, the function is onto.

3. Bijective or One-one and Onto Function: A bijective function is both one-one and onto. That means each input has its own unique output, and every output is used.

For example, if f(x) = x + 1 with domain {1, 2, 3} and codomain {2, 3, 4}, then each input has a unique output, and every element of the codomain is used. So, it’s bijective.

4. Invertible Function: An invertible function is a function where every output comes from exactly one input, so you can “reverse” it to get back the input.

For example, f(x) = x + 3.

If f(2) = 5, then the inverse function f^{-1}(x) = x - 3 gives f^{-1}(5) = 2.

For better practice, check out the CBSE Class 12 Maths Sample Papers.

Binary Operations in Class 12 Maths

A binary operation is just a function that takes two inputs from the same set and gives back one output in the same set.

For example, For set A = {1, 2, 3}, if we use addition (+), then 2 + 3 = 5. Since 5 is not in A, addition is not a binary operation on this set. But if A = integers, then addition is a binary operation, because the result always stays in the set.

1. Commutative Binary Operation

A commutative binary operation means the order of the two elements doesn’t matter — the result is the same.

For example, in addition, 2 + 3 = 3 + 2 = 5.

2. Associative Binary Operation

An associative binary operation means the way you group the elements doesn’t change the result.

For example, In addition, (2 + 3) + 4 = 2 + (3 + 4) = 9.

Conclusion

Relations show how elements of two sets connect. One input can link to many outputs, like a student joining different clubs. Functions are stricter; each input has exactly one output. For example, every student has just one roll number.

In short, all functions are relations, but not all relations are functions.

FAQs

Q1. What is the formula for a relation?

Ans. There is no single "formula" for relations. Relations are just rules that connect elements of sets, and we describe them based on how they behave, whether they loop back to themselves, work both ways, or link in chains.

Q2. What is the range of a relation?

Ans. The range of a relation is the set of all the possible output values, or the y-values, from the ordered pairs in that relation. 

Q3. What is the rule of relations?

Ans. A relation is a rule used to describe how one element of the domain is related to an element of the range.

Q4. What are the types of functions in Class 12 Relations and Functions?

Ans. Functions can be one-one, many-one, onto, bijective, and may also include special types like identity, constant, modulus, and signum.

Q5. What is the difference between relation and function Class 12?

Ans. A relation can map one input to multiple outputs, but a function maps every input to exactly one output.