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Question
Let f : N → N be defined by f(n) = `{((n+1)/2", if n is odd"),(n/2", if n is even"):}` for all n ∈ N.
State whether the function f is bijective. Justify your answer.
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Solution
f : N → N is defined as f(n) = `{((n + 1)/2",", "if n is odd"), (n/2",", "if n is even"):}` for all n ∈ N.
It can be observed that:
f(1) = `(1+1)/2` = 1 and f(2) = `2/2` = 1 ...[By definition of f]
∴ f(1) = f(2), where 1 ≠ 2
∴ f is not one-one.
Consider a natural number (n) in co-domain N.
Case I: n is odd
∴ n = 2r + 1 for some r ∈ N. Then, there exists 4r + 1 ∈ N such that
f(4r + 1) = `(4r + 1 + 1)/2`
= 2r + 1
Case II: n is even
∴ n = 2r for some r ∈ N. Then, there exists 4r ∈ N such that
f(4r) = `(4r)/2`
= 2r
∴ f is onto.
Hence, f is not a bijective function.
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