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Injective, Surjective and Bijective Functions

Let us consider a function from A to B and understand the types of function:
Description: 9490.png
 
  1. General Function – We have already discussed this. Two or more elements of A can have same mapping in B.
     
    For example: y = f (x) = x2. In this function, two values of x will point to same value of y. (x = 2 or x = –2 both will give y = 4).
  2. Injective – In this type of functions, each element in A has its unique corresponding mapping in B. B may have some elements without any correspondence in A.
     
    For example: y = f (x) = 3x + 2

    In this case, you won’t have two values of A having same mapping in B. So “many to one” is not possible. (One to many is obviously not possible because it’s a function).
  3. Surjective – In this type of functions, each element in B should be mapped by one or more than one element in A.
     
    There won’t be an element in “B” left out.
     
    Surjective Functions are also known as Onto function.
  4. Bijective – It means both Injective and Surjective.
     
    So each element of B will be mapped by a unique element in A.
Example-1
Indentify what type of function is this (x to y)?
  1. Description: 9509.png
    (i) A general function
    (ii) Injective
    (iii) Surjective
    (iv) Bijective
  2. Description: 9526.png
    (i) A general function
    (ii) Injective
    (iii) Surjective
    (iv) Bijective
  3. Description: 9544.png
    (i) A general function
    (ii) Injective
    (iii) Surjective
    (iv) Bijective
Solution
  1. Bijective is the answer.
  2. A general function is the answer because many to one mapping is there.
  3. It cannot be Injective because two X are pointing to single Y. Since no Y is left out, hence Surjective is the answer.




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