Relations and Functions


   
 
Functions (or Mappings)
Let A and B be two non-empty sets.
 
A function f from A to B is an association of every element of A to an
 
unique element in B. We write this as f : A g B.
 
A is called the domain.
 
B is called the co-domain.
 
Examples:
 
i)
 
 
Because the element c A is not associated with an element of B.
 
 
ii)
 
 
Because the element a A has no unique association.
 
 
 
Hence y = f(x), y is called the image of x and x is called the pre-image of y.
 
Range of f
 
 
Note:
 
 
Example:
 
 
Range of f = {1,2,3,4}
 
Types of functions (Mappings)
 
One-one function (or an injective map)
 
A function f : A g B is called a 1-1 mapping, if every element of f(A) does not have more than one pre-image.
 
 
Examples:
 
i)
 
 
ii)
 
 
Many-one function
 
A function which is not 1-1.
 
Example:
 
 
Onto function (or surjective map)
 
Let f : A g B be a function, f is said to be an onto function if f(A) = B.
 
Examples:
 
i)
 
 
ii)
 
 
Bijection
 
Let f : A g B be a function, f is said to be a bijection if f is 1 - 1 and f is onto.
 
Example:
 
 
Constant function
 
A function f : A g B is said to be a constant function, if
 
 
Example:
 
 
Identity function
 
 
Note:
 
Identity function is denoted by IA.
 
Example:
 
 
Equality of two functions
 
Two functions f and g are said to be equal if
 
domain of f = domain of g
 
codomain of f = codomain of g and
 
 
Composite function
 
Let f : A g B and g : B g C be two functions.
 
Let h: A g C be function such that h(x) = g(f(x))
 
 
h is called the composite of f and g, denoted by h = g o f.
 
i.e., (g o f) (x) = g (f(x))
 
Note:
 
 
 
 
(iii) Domain of gof is Domain of f.
 
Example:
 
 
 
(gof) (x) = g (f(x)) = g (x+1) = (x+1)2
 
(fog) (x) = f (g(x) = f (x2) = x2 + 1
 
Inverse of an element
 
 
f-1(b) = {a|b = f (a)}.
 
Example:
 
 
f-1(1) = {a}
 
f-1(2) = {b,c}
 
 
 
Inverse function
 
 
 
 
Example:
 
 
 
Note:
(i) Inverse of a function f is unique.
 
(ii) f-1of = fof-1 = IA
 
(iii) (f-1)-1= f
 
(iv) (fog)-1= g-1of -1
 
  
     
   
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