CBSE Class 12 Mathematics Relations And Functions Notes Set B

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
 
· 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

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.

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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