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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.

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