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Question
Let f: N → N be defined by f(n) = `{((n+1)/2, ",if n is odd"),(n/2,",n is even"):}` for all n ∈ N.
State whether the function f is bijective. Justify your answer.
Solution
f: N → N is defined as f(n) =`{((n+1)/2, ",if n is odd"),(n/2,",n is even"):}`
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.
But f is onto because range of f= N
[∵ For any x EN, 2x ∈ N such that `f(2x) = ((2x)/2) = x`]
⇒ f is onto.
Hence, f is not bijective
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