Answer:
Let
x1 and x2 are two distinct elements of W.
Injective
:
Case
? I : When both x1 and x2 are even :
As
Case
? II : When both x1 and x2 are odd :
f(x1)
= x1 ? 1 and f(x2) = x2 ? 1
As
f(x1)
Case-II
: When x1 is even and x2is odd :
f(x1)
= x1 + 1 and f(x2) = x2 ? 1
As
Similarly
f(x1)
for
odd x1 and even x2
Therefore,
in all cases
Hence
f is one-one function.
Surjective
: Let y be any element of W then f(y ? 1) = y, if is odd and f(y + 1) = y,
if y is even
Therefore,
corresponding to every element y of
W,
there exists elements y ? 1 (or y + 1) of W and f :
Hence f is onto function.
Thus f is both one-one
and onto function.
is
invertible.
To find inverse of f :
As f(n - 1) = n, if n
is odd
And f(n + 1) = n, if n is
even
So, f?1(n) =
= f(n)
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