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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
Here f(1) = `(1+1)/2 = 1` and f(2) = `2/2` = 1 ...[By definition of f]
∴ f(1) = f(2) but 1 ≠ 2
⇒ f is not one-one.
But f is onto because the range of f = N ...`[∵ "For any x ∈ N, 2x ∈ N such that f(2x)" = (2x)/2 = x]`
⇒ f is onto.
Hence, f is not bijective.
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