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Question
Let f : N → N be defined by
`f(n) = { (n+ 1, if n is odd),( n-1 , if n is even):}`
Show that f is a bijection.
[CBSE 2012, NCERT]
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Solution
We have,
`f (n) = {(n + 1 , if n is odd),(n - 1, if n is even):}`
Injection test :
Case I: If n is odd,
Let x, y ∈ N such that f (x)=f (y)
As, f (x)=f (y)
⇒ x + 1= y + 1
⇒ x = y
Case II: If n is even,
Let x, y ∈ N such that f (x)=f (y)
As, f (x)=f (y)
⇒ x − 1 = y − 1
⇒ x = y
So, f is injective.
Surjection test:
Case I: If n is odd,
As, for every n ∈ N, there exists y = n − 1 in N such that
f (y) = f (n−1)=n −1+1= n
Case II: If n is even,
As, for every n ∈ N, there exists y = n + 1 in N such that f (y)=f (n+1)=n +1−1 = n
So, f is surjective.
So, f is a bijection.
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