Q. 233.6( 5 Votes )

Let f : N

Answer :

TIP: One – One Function: – A function is said to be a one – one functions or an injection if different elements of A have different images in B.


So, is One – One function


a≠b


f(a)≠f(b) for all


f(a) = f(b)


a = b for all


Onto Function: – A function is said to be a onto function or surjection if every element of A i.e, if f(A) = B or range of f is the co – domain of f.


So, is Surjection iff for each , there exists such that f(a) = b


Bijection Function: – A function is said to be a bijection function if it is one – one as well as onto function.


Now, suppose


f(n1) = f(n2)


If n1 is odd and n2 is even, then we have


n1 + 1 = n2 – 2


n2 – n1 = 2


Not possible


Suppose both n1 even and n2 is odd.


Then, f(n1) = f(n2)


n1 – 1 = n2 + 1


n1 – n2 = 2


Not possible


Therefore, both n1 and n2 must be either odd or even


Suppose both n1 and n2 are odd.


Then, f(n1) = f(n2)


n1 + 1 = n2 + 1


n1 = n2


Suppose both n1 and n2 are even.


Then, f(n1) = f(n2)


n1 – 1 = n2 – 1


n1 = n2


Then, f is One – One


Also, any odd number 2r + 1 in the co – domain N will have an even number as image in domain N which is


f(n) = 2r + 1


n – 1 = 2r + 1


n = 2r + 2


Any even number 2r in the co – domain N will have an odd number as image in domain N which is


f(n) = 2r


n + 1 = 2r


n = 2r – 1


Thus f is Onto function.


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