let f:n-n defined by f(x)= n+1/2, if n is odd
n/2, if n i even
state whether f is bijective ?
i know how to prove that f is 1-1 . i need help for if f is onto or not ..
f: N ® N is defined by
For n = 1, we have
For n = 2, we have
Thus, f (1) = f (2) for 1 ≠ 2.
So, f is not one-one as two distinct elements in the domain have the same image under function f.
Suppose n be an arbitrary element of N.
If n is odd natural number, then 2n –1 is also an odd natural number.
If n is even natural number, then 2n is also an even natural number.
Thus, for every n in the codomain of f, there exists its pre image in the domain N. Hence, the range of f is equal to the codomain of f. So, f is onto.
The function is not one-one but onto. Hence, f is not bijective.