CBSE Class 12 Mathematics Relations And Functions Notes Set B

Download CBSE Class 12 Mathematics Relations And Functions Notes Set B in PDF format. All Revision notes for Class 12 Mathematics have been designed as per the latest syllabus and updated chapters given in your textbook for Mathematics in Class 12. Our teachers have designed these concept notes for the benefit of Class 12 students. You should use these chapter wise notes for revision on daily basis. These study notes can also be used for learning each chapter and its important and difficult topics or revision just before your exams to help you get better scores in upcoming examinations, You can also use Printable notes for Class 12 Mathematics for faster revision of difficult topics and get higher rank. After reading these notes also refer to MCQ questions for Class 12 Mathematics given on studiestoday

Revision Notes for Class 12 Mathematics Chapter 1 Relations and Functions

Class 12 Mathematics students should refer to the following concepts and notes for Chapter 1 Relations and Functions in Class 12. These exam notes for Class 12 Mathematics will be very useful for upcoming class tests and examinations and help you to score good marks

Chapter 1 Relations and Functions Notes Class 12 Mathematics

 

 

Relations and Functions

Points to Remember

Key Concepts

1. A relation R between two non empty sets A and B is a subset of their Cartesian Product A ´ B. If A = B then relation R on A is a subset of A ´ A

2. If (a, b) belongs to R, then a is related to b, and written as a R b If (a,b) does not belongs to R then a R b.

3. Let R be a relation from A to B. Then Domain of RÌ A and Range of RÌ B co domain is either set B or any of its superset or subset containing range of R

4. A relation R in a set A is called empty relation, if no element of A is related to any element of A, i.e., R = fÌ A × A.

5. A relation R in a set A is called universal relation, if each element of A is related to every element of A, i.e., R = A × A.

6. A relation R in a set A is called

a. Reflexive, if (a, a) Î R, for every a Î A,

b. Symmetric, if (a1, a2) Î R implies that (a2, a1) Î R, for all a1, a2 Î A.

c. Transitive, if (a1, a2) Î R and (a2, a3) Î R implies that (a1, a3) Î R, or all a1, a2, a3 Î A.

7. A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.

8. The empty relation R on a non-empty set X (i.e. a R b is never true) is not an equivalence relation, because although it is vacuously symmetric and transitive, it is not reflexive (except when X is also mpty)

9. Given an arbitrary equivalence relation R in a set X, R divides X into mutually disjoint subsets i S called partitions or subdivisions of X satisfying: 

· All elements of i S are related to each other, for all i
· No element of i S is related to j S ,if i ¹ j
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· The subsets j S are called Equivalence classes.

10. A function from a non empty set A to another non empty set B is a correspondence or a rule which associates every element of A to a unique element of B written as f:A ® B s.t f(x) = y for all xÎA, yÎB. All functions are relations but converse is not true.

11. If f: A ® B is a function then set A is the domain, set B is co-domain and set {f(x):x Î A } is the range of f. Range is a subset of codomain.

12. f: A ® B is one-to-one if For all x, yÎ A f(x) = f(y) Þ x = y or x ¹ y Þ f(x) ¹ f(y) A one- one function is known as injection or an Injective Function. Otherwise, f is called many-one.

13. f: A ® B is an onto function ,if for each b ÎB  there is atleast  one a Î A such that f(a) = b i.e if every element in B is the image of some element in A, f is onto.

14. A function which is both one-one and onto is called a bijective function or a bijection.

15. For an onto function range = co-domain.

16. A one – one function defined from a finite set to itself is always onto but if the set is infinite then it is not the case.  

Composition of f and g is written as gof and not fog gof is defined if the range of f Ì  domain of f and fog is defined if range of g Ì  domain of f

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18. Composition of functions is not commutative in general fog(x) ≠ gof(x).Composition is associative If f: X → Y, g: Y → Z and h: Z → S are functions then ho(g o f)=(h o g)of

19. A function f: X → Y is defined to be invertible, if there exists a function g : Y → X such that gof = IX and fog = IY. The function g is called the inverse of f and is denoted by f –1

20. If f is invertible, then f must be one-one and onto and conversely, if f is one- one and onto, then f must be invertible.

21. If f:A → B and g: B →C are one-one and onto then gof: A → C is also one-one and onto. But If g o f is one –one then only f is one –one g may or may not be one-one. If g o f is onto then g is onto f may or may not be onto.

22. Let f: X → Y and g: Y → Z be two invertible functions. Then gof is also Invertible with (gof)–1 = f –1o g–1.

23. If f: R → R is invertible, f(x)=y, then 1 f → (y)=x and (f-1)-1 is the function f itself.

24. A binary operation * on a set A is a function from A X A to A.

25.Addition, subtraction and multiplication are binary operations on R, the set of real numbers. Division is not binary on R, however, division is a binary operation on R-{0}, the set of non-zero real numbers

26.A binary operation ∗ on the set X is called commutative, if a→ b= b→ a, for every a,b→X

27.A binary operation → on the set X is called associative, if a∗ → (b*c) =(a*b)*c, for every a, b, c→X

28.An element e ∈ A is called an identity of A with respect to *, if for each a ∈ A, a * e = a = e * a. The identity element of (A, *) if it exists, is unique.

29.Given a binary operation * from A x A → A, with the identity element e in A, an element a∈ A is said to be invertible with respect to the operation * , if there exists an element b in A such that a* b=e= b* a, then b is called the inverse of a and is denoted by a-1.

30.If the operation table is symmetric about the diagonal line then, the operation is commutative.

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31. Addition '+' and multiplication '·' on N, the set of natural numbers are binary operations But subtraction ‘–‘ and division are not since (4, 5) = 4 - 5 = -1 ∉ N and 4/5 =.8 ∉ N

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CBSE Class 12 Mathematics Chapter 1 Relations and Functions Notes

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Notes for Mathematics CBSE Class 12 Chapter 1 Relations and Functions

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Chapter 1 Relations and Functions CBSE Class 12 Mathematics Notes

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Notes for CBSE Mathematics Class 12 Chapter 1 Relations and Functions

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