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
⇒ 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, f : N → N given by f(x) = x3
Check for Injectivity:
Let x,y be elements belongs to N i.e such that
⇒ f(x) = f(y)
⇒ x3 = y3
⇒ x3 – y3 = 0
⇒ (x – y)(x2 + y2 + xy) = 0
As therefore x2 + y2 + xy >0
⇒ x – y = 0
⇒ x = y
Hence f is One – One function
Check for Surjectivity:
Let y be element belongs to N i.e be arbitrary, then
⇒ f(x) = y
⇒ x3 = y
⇒ not belongs to N for non–perfect cube value of y.
Since f attain only cubic number like 1,8,27….,
Therefore no non – perfect cubic values of y in N (co – domain) has a pre–image in domain N.
Hence, f is not onto function
Thus, Not Bijective also
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