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Let F: N → N Be Defined by F(N) State Whether the Function F is Bijective. Justify Your Answer. - Mathematics

Let fN → N be defined by f(n) = `{((n+1)/2, "if n is odd"),(,"   for all n ∈ N"), (n/2, if "n is even"):}`

State whether the function f is bijective. Justify your answer.

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Solution

fN → N is defined as f(n) = `{((n+1)/2, "if n is odd"),(,"   for all n ∈ N"), (n/2, if "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.

Consider a natural number (n) in co-domain N.

Case I: n is odd

n = 2r + 1 for some r ∈ N. Then, there exists 4+ 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.

  Is there an error in this question or solution?
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APPEARS IN

NCERT Class 12 Maths
Chapter 1 Relations and Functions
Q 9 | Page 11
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