Class notes Mathematics

FORMULA LIST
Relations and Functions
(1) Relation: Let A and B be two sets. Then a relation R from set A to set B is a subset of A B .
(2) Types of Relations:
2a. Empty Relation: A relation R on a set A is said to an empty relation iff R   i.e.

 a, b   R a, b  A i.e. no element of A will be related to any other element of A with the help of

the relation R.
2b. Universal Relation: A relation R on a set A is said to be a universal relation iff
R  A  A i.e.  a, b   R, a,b  A i.e. each element of A is related to every other element of A with




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the help of the relation R.




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(3) A relation R on A is:

 a , a  R for all a  A



.
3a. Reflexive Relation if

3b. Symmetric Relation if  hs
a , b  R   b , a   R all a , b  A
at
3c. Transitive Relation if 
a , b   R ,  b , c   R   a , c   R all a , b , c  A
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(4) A relation R in a set A is said to be an Equivalence Relation if R is reflexive, symmetric and
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transitive.
(5) Equivalence Class
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Let R be an equivalence relation on a set A and let a  A . Then, we define the equivalence class of
itb




a as  a   b  A, b is related to a  b A :  b, a   R
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(6) Function: Let A and B be two non-empty sets. Then a function ‘ f ’ from set A to set B is a
rule which associates elements of set A to elements of set B such that:
(a) All elements of set A are associated to elements in set .
(b) An element of set A is associated to a unique element in set B .
(7) Vertical Line Test: A curve in a plane represents the graph of a real function if and only if no
vertical line intersects it more than once.




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,(8) Types of functions:
8a. One-One Function (or Injective Function): A function y  f  x  is said to be one –

one iff different pre-images have different images or if images are same
then the pre-images are also same.
i.e. f  x1   f  x2   x1  x2 or x1  x2  f  x1   f  x2 

8b. Many-One Function: A function in which at least two pre-
images have same image is called as many-one function.




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8c. Into Function: A Function is said to be an ‘into’ function if there is at least one element
in the co-domain of the function such that it has no pre-image in domain.




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8d. Onto Function (or Surjective Function): A function is said to be onto function if each




.
element of co-domain has a pre-image in its domain. Alternately, A function ‘f’ will be called an
onto function iff R f  co-domain  f  hs
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8e. Bijective Function: A function which is one – one and onto both is called as bijective
function.
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(9) Horizontal Line Test: f is one-one function if no line parallel to x-axis meets the graph in more
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than one point.
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(10) Let A be any finite set having n elements. Then,
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10a. Number of one-one functions from A to A are n !
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10b. Number of onto functions from A to A are n !

10c. Number of bijective functions from A to A are n !

(11) If A and B are finite sets containing m and n elements, then

11a. Total number of relations from the set A to set B is 2 mn .
2
11b. Total number of relations on the set A is 2 m .

11c. Total number of functions from the set A to set B is n m .

11d. Total number of one-one functions from the set A to set B is n Pm if n  m , otherwise 0.




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, n
nr n
11e. Total number of onto functions from set A to set B is  (1) Cr r m if m  n ,
r 1



otherwise 0.

11f. Total number of bijective functions from the set A to set B is m! , if m  n , otherwise 0.

(12) COMPOSITION OF FUNCTIONS
If f and g are two functions then their composition

12a. fog is defined iff Rg  D f and fog  x   f  g  x  

12b. gof is defined iff R f  Dg and gof  x   g  f  x  




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(13) a. If f : A  B and g : B  C then gof : A  C .




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b. If f : A  B and g : B  A then fog and gof are both defined, where gof : A  A and
fog : B  B




.
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(14) Identity Function: A function ‘I’ on a set A is said to be an identity function iff I : A  A ,
I  x  x .
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(15) Equal Functions: A function f will be equal to another function g iff
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(i) D f  Dg and (ii) f  x   g  x   x  D f  or Dg 
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(16) Invertible Functions: A function f : A  B is said to be an invertible function iff there exist
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another function g : B  A such that fog  I B and gof  I A . And we write g  f 1
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(17) a. A function is invertible iff it is one-one and onto.
b. In the above definition f and g are both inverse of each other i.e. f 1  g and g 1  f .
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1
c.  f 1   f .
1
d.  fog   g 1of 1

e. fof 1  f 1of  I




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, Trigonometry and Inverse Trigonometry
(1)1800   radians
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(2)   ,  is measured in radians
r
(3) Trigonometric Ratios of Special Angles
1 1 3
cos 45  cos 60  cos 30 
2 2
2
1
1 3 sin 30 
sin 45  sin 60  2
2 2
1
tan 45  1 tan 60  3 tan 30 
3
(4) sin  90     cos cos  90     sin




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;
tan  90     cot  ;  
cot 900   tan 




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sec  90     cos ec ; cosec  90     sec 




.
(5) cos     cos  ; sin      sin  ; tan      tan 

(6) sec  
1
; cosec 
1
; cot  
1
; tan  
hs
sin 
; cot  
cos 
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cos  sin tan  cos  sin 
(7)sin 2   cos 2   1 ; 1  tan 2   sec 2  ; 1  cot 2   cos ec 2
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(8)  1  sin x  1 ;  1  cos x  1 ;    tan x  
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(9) sin  A  B   sin A cos B  cos A sin B
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(10) sin  A  B   sin A cos B  cos A sin B
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(11) cos  A  B   cos A cos B  sin A sin B
(12) cos  A  B   cos A cos B  sin A sin B
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tan A  tan B
(13) tan  A  B  
1  tan A tan B
tan A  tan B
(14) tan  A  B  
1  tan A tan B
(15) 2 sin A cos B  sin  A  B   sin  A  B 
(16) 2 cos A sin B  sin  A  B   sin  A  B 
(17) 2 cos A cos B  cos  A  B   cos  A  B 
(18) 2 sin A sin B  cos  A  B   cos  A  B 




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