Personal Resilience in Project Management 2 - TV Edit 1a.pdf
MB1201 QR
1. In this document you can review important
• Definitions
• Facts
• Formulae
• Procedures
Class – XII, CBSE
MB1201: Relations And
Functions
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2. Define a relation R
from a set A to set B.
Class – XII,CBSE MB1201: Relations And Functions
Important Definitions
Topic: Relations
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3. Define a relation R
from a set A to set B.
Important Definitions
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Topic: Relations
A relation R from a set A to set B is
a subset of A × B . If R is a relation
from a set A to set B and, if
, , then w e say that is
related to under the relation R
and w e express this fact by
w ri
a b R a
b
ting as R .a b
Class – XII,CBSE MB1201: Relations And Functions
4. Define a relation R
in set A.
Important Definitions
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Topic: RelationsClass – XII,CBSE MB1201: Relations And Functions
5. Define a relation R
in set A.
Important Definitions
A relation R in a set A is a subset of .A A
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Topic: RelationsClass – XII,CBSE MB1201: Relations And Functions
6. What is empty
relation?
Important Definitions
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Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions
7. What is empty
relation?
Important Definitions
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Topic: Types of Relations
A relation R in a set A is called
, if no element of A is related
to any element of A, i.e., .
empty
relation
R A A
Class – XII,CBSE MB1201: Relations And Functions
8. What is universal
relation?
Important Definitions
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Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions
9. What is universal
relation?
Important Definitions
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Topic: Types of Relations
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.
Both the empty relation and the
universal relation are called as
trivial relations.
Class – XII,CBSE MB1201: Relations And Functions
10. Define different
types of relations in
a set.
Important Definitions
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Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions
11. Define different
types of relations in
a set.
Important Definitions
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Topic: Types of Relations
1 2
2 1 1 2
1 2
2 3 1 3
1 2
A relation R in a set A is called:
Reflexive, if , , for every
,
Symmetric, if , implies
that , , for all , .
Transitive, if , and
, implies that , ,
for all , ,
a a R
a A
a a R
a a R a a R
a a R
a a R a a R
a a a3
.A
Class – XII,CBSE MB1201: Relations And Functions
12. What is equivalence
relation?
Important Definitions
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Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions
13. What is equivalence
relation?
Important Definitions
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Topic: Types of Relations
A relation R in a set A is said to be an
if R is reflexive,
symmetric and transitive.
equivalence relation
Class – XII,CBSE MB1201: Relations And Functions
14. Define one-one and
many-one function.
Important Definitions
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Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions
15. Define one-one and
many-one function.
Important Definitions
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Topic: Types of Functions
1 2
1 2 1 2
A function : is defined to be
one-one (or injective), if the images
of distinct elements of X under are
distinct, i.e., for every , ,
implies .
Otherwise, is called many-one.
f X Y
f
x x X
f x f x x x
f
Class – XII,CBSE MB1201: Relations And Functions
16. Define one-one and
many-one function.
Important Definitions
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Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions
1 4
2 3
The function and in the above
Figure (i) and (iv) are one-one and
the function and in Figure (ii)
and (iii) are many-one.
f f
f f
17. Define onto function.
Important Definitions
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Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions
18. Define onto function.
Important Definitions
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Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions
A function : is said to be
( ), if every element of Y
is the image of some element of X
under , i.e., for every , there exists
an element in X such that .
f X Y onto
or surjective
f y Y
x f x y
19. Define onto function.
Important Definitions
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Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions
3 4
1
2
1 1
The function and in Figure (iii), (iv)
are onto and the function in Figure (i)
is not onto as elements and in are
not the image of any element in under .
f f
f
e f X
X f
21. Define bijective
function.
Important Definitions
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Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions
A function : is said to be
one-one and onto (or bijective), if
is both one-one and onto.
f X Y
f
4
T he function in Figure (iv) is one-one
and onto.
f
22. Define composition
of functions.
Important Definitions
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Topic: Composition of FunctionsClass – XII,CBSE MB1201: Relations And Functions
23. Define composition
of functions.
Important Definitions
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Topic: Composition of FunctionsClass – XII,CBSE MB1201: Relations And Functions
Let : and : be two
functions. Then the composition of
and , denoted by , is defined
as the function : given by
, .
f A B g B C
f g g f
g f A C
g f x g f x x A
24. Define composition
of functions.
Important Definitions
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Topic: Composition of FunctionsClass – XII,CBSE MB1201: Relations And Functions
25. What is invertible
function?
Important Definitions
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Topic: Invertible FunctionClass – XII,CBSE MB1201: Relations And Functions
26. What is invertible
function?
Important Definitions
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Topic: Invertible FunctionClass – XII,CBSE MB1201: Relations And Functions
1
A function : is said to be inver-
tible, if there exists a function :
such that and . The
function is called the and is
denoted by , here d
X Y
X
f X Y
g Y X
g f I f g I
g inverse of f
f I
Invertible function :
enotes the identity
function on the set X ,and denotes the
identity function on the set Y.
Y
I
27. Define different
binary operations
on a set.
Important Definitions
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Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions
28. Define different
binary operations
on a set.
Important Definitions
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Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions
A binary operation on a set A is a
function : . W e denote ( , ),
the value of the function on the element
(a, b) by .
A binary operation on the set A is
called commutative, if ,
for eve
A A A a b
a b
a b b a
ry , .a b A
29. Define different
binary operations
on a set.
Important Definitions
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Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions
A binary operation on the set a is
said to be associative if,
, , , , .
Given a binary operation :
an element , if it exists, is called
identity for the operation , if
,
a b c a b c a b c A
A A A
e A
a e a e a a .A
30. Define different
binary operations
on a set.
Important Definitions
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Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions
Given a binary operation :
with the identity element in A, an
element is said to be with
respect to the operation , if there exists an
element in A such that
and
A A A
e
a A invertible
b a b e b a
b
1
is called the inverse and is
denoted by .
of a
a
31. Important Facts
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Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions
.R X X
Fact # 1
Empty relation is the relation R in a set X given by
32. Important Facts
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Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions
.R X X
Fact # 2
Universal relation is the relation R in a set X given by
33. Important Facts
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Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions
, .a a R a X
Fact # 3
Reflexive relation R in X is a relation with
34. Important Facts
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Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions
,a b R , .b a R
Fact # 4
Symmetric relation R in X is a relation satisfying implies
35. Important Facts
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Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions
,a b R ,b a R
, .a c R
Fact # 5
Transitive relation R in X is a relation satisfying and
implies that
36. Important Facts
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Topic: Equivalence RelationsClass – XII,CBSE MB1201: Relations And Functions
Fact # 6
Equivalence relation R in X is a relation which is reflexive, symmetric
and transitive.
37. Important Facts
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Topic: Equivalence RelationsClass – XII,CBSE MB1201: Relations And Functions
a a X
Fact # 7
Equivalence class containing for an equivalence relation R in a
set X is the subset of X containing all elements b related to a.
38. Important Facts
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Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions
:f X Y
1 2 1 2 1 2
, .f x f x x x x x X
Fact # 8
A function is one-one (or injective) if
39. Important Facts
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Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions
:f X Y ,y Y x X
.f x y
Fact # 9
A function is onto (or surjective) if given any
such that
40. Important Facts
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Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions
:f X Y
Fact # 10
A function is one-one and onto (or bijective), if f is both
one-one and onto.
41. Important Facts
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Topic: Composition of FunctionsClass – XII,CBSE MB1201: Relations And Functions
:f A B :g B C
:g f A C .gof x g f x x A
Fact # 11
The composition of functions and is the function
given by
42. Important Facts
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Topic: Invertible FunctionsClass – XII,CBSE MB1201: Relations And Functions
:f X Y :g Y X X
gof I
.Y
fog I
Fact # 12
A function is invertible if such that and
43. A function is invertible if and only if f is one-one and onto.
That is “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”. This fact significantly
helps for proving a function f to be invertible by showing that f is
one-one and onto, specially when the actual inverse of f is not to be
determined.
Important Facts
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Topic: Invertible FunctionsClass – XII,CBSE MB1201: Relations And Functions
:f X Y
Fact # 13
44. Important Facts
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Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions
:f X X
Fact # 14
Given a finite set X, a function is one-one (respectively onto) if
and only if f is onto (respectively one-one). This is the characteristic
property of a finite set. This is not true for infinite set.
45. Important Facts
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Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions
A A
Fact # 15
A binary operation on a set A is a function from to A.
46. Important Facts
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Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions
e X : ,X X
.a e a e a X
Fact # 16
An element is the identity element for binary operation
If
47. An element is invertible for binary operation if
there exists such that where, e is the identity for
the binary operation
The element b is called inverse of and is denoted by .
Important Facts
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Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions
a X : ,X X X
b X a b e b a
.
1
a
Fact # 17
a
48. Important Facts
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Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions
,a b b a a b
Fact # 18
An operation on X is commutative if in X.
49. Important Facts
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Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions
An operation on X is associative if , , in X.a b c a b c a b c
Fact # 19
50. Fact # 20
Important Facts
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Topic: Equivalence RelationsClass – XII,CBSE MB1201: Relations And Functions
Given an arbitrary equivalence relation R in an arbitrary set X, R divides X
into mutually disjoint subsets called partitions or subdivisions of X
satisfying the following:
All elements of are re
i
i
A
A lated to each other, for all .
No element of is related to any element of , .i j
i
A A i j
51. Fact # 20
Important Facts
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Topic: Equivalence ClassesClass – XII,CBSE MB1201: Relations And Functions
and , .
The subsets are called .
j i j
i
A X A A i j
A equivalence classes
52. Fact # 21
Important Facts
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Topic: Equivalence RelationsClass – XII,CBSE MB1201: Relations And Functions
If R and S are two equivalence relations on a set A, then is also an
equivalence relation on A.
R S
53. Fact # 22
Important Facts
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Topic: Equivalence RelationsClass – XII,CBSE MB1201: Relations And Functions
The union of two equivalence relations on a set is not necessarily an
equivalence relation on the set.
54. Fact # 23
Important Facts
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Topic: Equivalence RelationsClass – XII,CBSE MB1201: Relations And Functions
1
1
If R is an equivalence relation on a set A, then defined as
, : , is also an equivalence relation on A .
R
R b a a b R
55. Fact # 24
Important Facts
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Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions
For an arbitrary finite set X, a one-one function : is necessarily
onto and an onto map : is necessarily one-one. For an infinite
set, this may not be true. In fact, this is a characteristic d
f X X
f X X
ifference between
a finite and an infinite set.
56. Fact # 25
Important Facts
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Topic: Identity Element for the OperationsClass – XII,CBSE MB1201: Relations And Functions
Zero is the identity for the addition operation on R but it is not identity
for the addition operation on N, as 0 . In fact the addition operation
on N does not have any identity.
N
57. Fact # 26
Important Facts
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Topic: Identity Element for the OperationsClass – XII,CBSE MB1201: Relations And Functions
For the addition operation : , given any , there exists
in R such that a + = 0 (identity for '+') .
a
a a a a
R R R R
58. Fact # 27
Important Facts
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Topic: Identity Element for the OperationsClass – XII,CBSE MB1201: Relations And Functions
For the multiplication operation on R, given any 0 in R, we can choose
1 1 1
in R such that 1 (identity for '×') = .
a
a a
a a a
59. Important Formulae
If : , : and are functions, then
.
f X Y g Y Z Z S
h g f h g f
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Topic: Composition of FunctionsClass – XII,CBSE MB1201: Relations And Functions
60. Important Formulae
1 1 1
If : and : be two invertible functions. Then is also
invertible with .
f X Y g Y Z g f
g f f g
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Topic: Composition of FunctionsClass – XII,CBSE MB1201: Relations And Functions
61. Important Procedures
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Topic: RelationsClass – XII,CBSE MB1201: Relations And Functions
Let A be the set of all straight lines in the plane. Define a relation 'R' on A.
62. Important Procedures
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Topic: RelationsClass – XII,CBSE
If A is the set of all straight lines in the plane.
A relation 'R' on A can be defined as follows:
or , if and only if a is parallel to be.
W e can easily verify that this relation is an equivale
aRb a b R
nce relation on A.
Thus, we can define its equivalence class denoted by as follows:
: , .
a A a
a b A a b R
MB1201: Relations And Functions
Let A be the set of all straight lines in the plane. Define a relation 'R' on A.
63. Important Procedures
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Topic: RelationsClass – XII,CBSE
W e note the following
, , or .
For all , , the union of all equivalence classes of elements of
A is equal to A.
a a A
a b A a b a b
a A a
MB1201: Relations And Functions
Let A be the set of all straight lines in the plane. Define a relation 'R' on A.
64. How to use operation table for the binary operation?
Important Procedures
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Topic: Operation Table for the OperationClass – XII,CBSE MB1201: Relations And Functions
65. How to use operation table for the binary operation?
Important Procedures
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Topic: Operation Table for the OperationClass – XII,CBSE
W hen number of elements in a set A is sm all, we can express a binary
operation on the set A with the help of a table called the operation table
for the operation . For example consider A = {1, 2, 3}. Then, the operation
on A defined as : given by ( , ) max { , } can be expressed
by the following operation table Here, (1, 3) = 3, (2, 3) = 3, (1, 2) = 2.
A A A a b a b
MB1201: Relations And Functions
3* 1 2
31 1 2
32 2 2
3 3 3 3