The document discusses important concepts related to relations and functions. It defines what a relation is and different types of relations such as reflexive, symmetric, transitive, and equivalence relations. It also defines different types of functions including one-to-one, onto, bijective, and inverse functions. It provides examples of binary operations and discusses their properties like commutativity, associativity, and identity elements. It concludes with short answer and very short answer type questions related to these concepts.
Analytic Function, C-R equation, Harmonic function, laplace equation, Construction of analytic function, Critical point, Invariant point , Bilinear Transformation
1. understand the terms function, domain, range, one-one function,inverse function and composition of functions
2. identify the range of a given function in simple cases, and find the
composition of two given functions
3. determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases
4. illustrate in graphical terms the relation between a one-one function and its inverse.
Chapter wise important questions in Mathematics for Karnataka 2 year PU Science students. This is taken from the PU board website and compiled together.
Analytic Function, C-R equation, Harmonic function, laplace equation, Construction of analytic function, Critical point, Invariant point , Bilinear Transformation
1. understand the terms function, domain, range, one-one function,inverse function and composition of functions
2. identify the range of a given function in simple cases, and find the
composition of two given functions
3. determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases
4. illustrate in graphical terms the relation between a one-one function and its inverse.
Chapter wise important questions in Mathematics for Karnataka 2 year PU Science students. This is taken from the PU board website and compiled together.
(a) Natural Numbers : N = {1,2,3,4,...}
(b) Whole Numbers : W = {0,1,2,3,4, }
(c) Integer Numbers :
or Z = {...–3,–2,–1, 0,1,2,3, },
Z+ = {1,2,3,....}, Z– = {–1,–2,–3, }
Z0 = {± 1, ± 2, ± 3, }
(d) Rational Numbers :
p
Q = { q ; p, q z, q 0 }
(i) R0 : all real numbers except 0 (Zero).
(j) Imaginary Numbers : C = {i,, }
(k) Prime Numbers :
These are the natural numbers greater than 1 which is divisible by 1 and itself only, called prime numbers.
Ex. 2,3,5,7,11,13,17,19,23,29,31,37,41,...
(l) Even Numbers : E = {0,2,4,6, }
(m) Odd Numbers : O = {1,3,5,7, }
Ex. {1,
Note :
5
, –10, 105,
3
22 20
7 , 3
, 0 ....}
The set of the numbers between any two real numbers is called interval.
(a) Close Interval :
(i) In rational numbers the digits are repeated after decimal.
(ii) 0 (zero) is a rational number.
(e) Irrational numbers: The numbers which are not rational or which can not be written in the form of p/q ,called irrational numbers
Ex. { , ,21/3, 51/4, ,e, }
Note:
(i) In irrational numbers, digits are not repeated after decimal.
(ii) and e are called special irrational quantities.
(iii) is neither a rational number nor a irrational number.
(f) Real Numbers : {x, where x is rational and irrational number}
20
[a,b] = { x, a x b }
(b) Open Interval:
(a, b) or ]a, b[ = { x, a < x < b }
(c) Semi open or semi close interval:
[a,b[ or [a,b) = {x; a x < b}
]a,b] or (a,b] = {x ; a < x b}
Let A and B be two given sets and if each element a A is associated with a unique element b B under a rule f , then this relation is called function.
Here b, is called the image of a and a is called the pre- image of b under f.
Note :
(i) Every element of A should be associated with
Ex. R = { 1,1000, 20/6, ,
, –10, –
,.....}
3
B but vice-versa is not essential.
(g) Positive Real Numbers: R+ = (0,)
(h) Negative Real Numbers : R– = (– ,0)
(ii) Every element of A should be associated with a unique (one and only one) element of but
any element of B can have two or more rela- tions in A.
3.1 Representation of Function :
It can be done by three methods :
(a) By Mapping
(b) By Algebraic Method
(c) In the form of Ordered pairs
(A) Mapping :
It shows the graphical aspect of the relation of the elements of A with the elements of B .
Ex. f1:
f2 :
f3 :
f4 :
In the above given mappings rule f1 and f2
shows a function because each element of A is
associated with a unique element of B. Whereas
f3 and f4 are not function because in f 3, element c is associated with two elements of B, and in f4 , b is not associated with any element
of B, which do not follow the definition of function. In f2, c and d are associated with same element, still it obeys the rule of definition of function because it does not tell that every element of A should be associated with different elements of B.
(B) Algebraic Method :
It shows the relation between the elem
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
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Relations and functions
1. 9 XII – Maths
CHAPTER 1
RELATIONS AND FUNCTIONS
IMPORTANT POINTS TO REMEMBER
Relation R from a set A to a set B is subset of A × B.
A × B = {(a, b) : a A, b B}.
If n(A) = r, n (B) = s from set A to set B then n (A × B) = rs.
and no. of relations = 2rs
is also a relation defined on set A, called the void (empty) relation.
R = A × A is called universal relation.
Reflexive Relation : Relation R defined on set A is said to be reflexive
iff (a, a) R a A
Symmetric Relation : Relation R defined on set A is said to be symmetric
iff (a, b) R (b, a) R a, b, A
Transitive Relation : Relation R defined on set A is said to be transitive
if (a, b) R, (b, c) R (a, c) R a, b, c R
Equivalence Relation : A relation defined on set A is said to be
equivalence relation iff it is reflexive, symmetric and transitive.
One-One Function : f : A B is said to be one-one if distinct elements
in A has distinct images in B. i.e. x1, x2 A s.t. x1 x2 f(x1) f(x2).
OR
x1, x2 A s.t. f(x1) = f(x2)
x1 = x2
One-one function is also called injective function.
2. XII – Maths 10
Onto function (surjective) : A function f : A B is said to be onto iff
Rf = B i.e. b B, there exist a A s.t. f(a) = b
A function which is not one-one is called many-one function.
A function which is not onto is called into.
Bijective Function : A function which is both injective and surjective is
called bijective.
Composition of Two Function : If f : A B, g : B C are two
functions, then composition of f and g denoted by gof is a function from
A to C given by, (gof) (x) = g (f (x)) x A
Clearly gof is defined if Range of f domain of g. Similarly fog can be
defined.
Invertible Function : A function f : X Y is invertible iff it is bijective.
If f : X Y is bijective function, then function g : Y X is said to be
inverse of f iff fog = Iy and gof = Ix
when Ix, Iy are identity functions.
g is inverse of f and is denoted by f –1.
Binary Operation : A binary operation ‘*’ defined on set A is a function
from A × A A. * (a, b) is denoted by a * b.
Binary operation * defined on set A is said to be commutative iff
a * b = b * a a, b A.
Binary operation * defined on set A is called associative iff a * (b * c)
= (a * b) * c a, b, c A
If * is Binary operation on A, then an element e A is said to be the
identity element iff a * e = e * a a A
Identity element is unique.
If * is Binary operation on set A, then an element b is said to be inverse
of a A iff a * b = b * a = e
Inverse of an element, if it exists, is unique.
3. 11 XII – Maths
VERY SHORT ANSWER TYPE QUESTIONS (1 MARK)
1. If A is the set of students of a school then write, which of following
relations are. (Universal, Empty or neither of the two).
R1 = {(a, b) : a, b are ages of students and |a – b| 0}
R2 = {(a, b) : a, b are weights of students, and |a – b| < 0}
R3 = {(a, b) : a, b are students studying in same class}
2. Is the relation R in the set A = {1, 2, 3, 4, 5} defined as R = {(a, b) : b
= a + 1} reflexive?
3. If R, is a relation in set N given by
R = {(a, b) : a = b – 3, b > 5},
then does elements (5, 7) R?
4. If f : {1, 3} {1, 2, 5} and g : {1, 2, 5} {1, 2, 3, 4} be given by
f = {(1, 2), (3, 5)}, g = {(1, 3), (2, 3), (5, 1)}
Write down gof.
5. Let g, f : R R be defined by
2
, 3 2. Write fog.
3
x
g x f x x
6. If f : R R defined by
2 1
5
x
f x
be an invertible function, write f–1(x).
7. If 1, Write .
1
x
f x x fo f x
x
8. Let * is a Binary operation defined on R, then if
(i) a * b = a + b + ab, write 3 * 2
4. XII – Maths 12
(ii)
2
* , Write 2 * 3 * 4.
3
a b
a b
9. If n(A) = n(B) = 3, Then how many bijective functions from A to B can be
formed?
10. If f (x) = x + 1, g(x) = x – 1, Then (gof) (3) = ?
11. Is f : N N given by f(x) = x2 is one-one? Give reason.
12. If f : R A, given by
f(x) = x2 – 2x + 2 is onto function, find set A.
13. If f : A B is bijective function such that n (A) = 10, then n (B) = ?
14. If n(A) = 5, then write the number of one-one functions from A to A.
15. R = {(a, b) : a, b N, a b and a divides b}. Is R reflexive? Give reason?
16. Is f : R R, given by f(x) = |x – 1| is one-one? Give reason?
17. f : R B given by f(x) = sin x is onto function, then write set B.
18. If
2
1 x 2x
f x log , show that f 2f x .
1 x 1 x
19. If ‘*’ is a binary operation on set Q of rational numbers given by *
5
ab
a b
then write the identity element in Q.
20. If * is Binary operation on N defined by a * b = a + ab a, b N. Write
the identity element in N if it exists.
SHORT ANSWER TYPE QUESTIONS (4 Marks)
21. Check the following functions for one-one and onto.
(a)
2 3
: , ( )
7
x
f R R f x
(b) f : R R, f(x) = |x + 1|
(c) f : R – {2} R,
3 1
2
x
f x
x
5. 13 XII – Maths
(d) f : R [–1, 1], f(x) = sin2x
22. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by
a* b = H.C.F. of a and b. Write the operation table for the operation *.
23. Let
4 4
:
3 3
f R R be a function given by
4
.
3 4
x
f x
x
Show
that f is invertible with
1 4
.
4 3
x
f x
x
24. Let R be the relation on set A = {x : x Z, 0 x 10} given by
R = {(a, b) : (a – b) is multiple of 4}, is an equivalence relation. Also, write
all elements related to 4.
25. Show that function f : A B defined as 3 4
5 7
x
f x
x
where
7 3
,
5 5
A R B R is invertible and hence find f –1.
26. Let * be a binary operation on Q. Such that a * b = a + b – ab.
(i) Prove that * is commutative and associative.
(ii) Find identify element of * in Q (if it exists).
27. If * is a binary operation defined on R – {0} defined by 2
2
* ,
a
a b
b
then
check * for commutativity and associativity.
28. If A = N × N and binary operation * is defined on A as
(a, b) * (c, d) = (ac, bd).
(i) Check * for commutativity and associativity.
(ii) Find the identity element for * in A (If it exists).
29. Show that the relation R defined by (a, b) R(c, d) a + d = b + c on
the set N × N is an equivalence relation.
30. Let * be a binary operation on set Q defined by * ,
4
ab
a b show that
(i) 4 is the identity element of * on Q.
6. XII – Maths 14
(ii) Every non zero element of Q is invertible with
1 16
, 0 .
a a Q
a
31. Show that f : R+ R+ defined by
1
2
f x
x
is bijective where R+ is the
set of all non-zero positive real numbers.
32. Consider f : R+ [–5, ) given by f(x) = 9x2 + 6x – 5 show that f is
invertible with
1 6 1
3
x
f
.
33. If ‘*’ is a binary operation on R defined by a * b = a + b + ab. Prove that
* is commutative and associative. Find the identify element. Also show
that every element of R is invertible except –1.
34. If f, g : R R defined by f(x) = x2 – x and g(x) = x + 1 find (fog) (x) and
(gof) (x). Are they equal?
35. 11
: [1, ) [2, ) is given by , find .
f f x x f x
x
36. f : R R, g : R R given by f(x) = [x], g(x) = |x| then find
2 2
and .
3 3
fog gof
ANSWERS
1. R1 : is universal relation.
R2 : is empty relation.
R3 : is neither universal nor empty.
2. No, R is not reflexive.
3. (5, 7) R
4. gof = {(1, 3), (3, 1)}
5. (fog)(x) = x x R
7. 15 XII – Maths
6. 1 5 1
2
x
f x
7. 1
,
2 1 2
x
fof x x
x
8. (i) 3 * 2 = 11
(ii)
1369
27
9. 6
10. 3
11. Yes, f is one-one 2 2
1 2 1 2,x x N x x .
12. A = [1, ) because Rf = [1, )
13. n(B) = 10
14. 120.
15. No, R is not reflexive ,a a R a N
16. f is not one-one functions
f(3) = f (–1) = 2
3 – 1 i.e. distinct element has same images.
17. B = [–1, 1]
19. e = 5
20. Identity element does not exist.
21. (a) Bijective
(b) Neither one-one nor onto.
(c) One-one, but not onto.
(d) Neither one-one nor onto.
8. XII – Maths 16
22.
* 1 2 3 4 5
1 1 1 1 1 1
2 1 2 1 2 1
3 1 1 3 1 1
4 1 2 1 4 1
5 1 1 1 1 5
24. Elements related to 4 are 0, 4, 8.
25. 1 7 4
5 3
x
f x
x
26. 0 is the identity element.
27. Neither commutative nor associative.
28. (i) Commutative and associative.
(ii) (1, 1) is identity in N × N
33. 0 is the identity element.
34. (fog) (x) = x2 + x
(gof) (x) = x2 – x + 1
Clearly, they are unequal.
35.
2
1 4
2
x x
f x
36.
2
0
3
fog
2
1
3
gof