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KCET2014MathematicsDifferential Equations

Let ( f: N N ) defined by ( f(n)= array cl n+1 2 ; & if n is odd n 2 ; & if n is even array . ) then ( f ) is

Options

  1. Aone-one and onto
  2. Bone-one but not onto
  3. Conto but not one-one
  4. Dneither one-one nor onto

Correct answer

C. onto but not one-one

Step-by-step solution

Given that, ( f: N N ) defined by ( f(n)= array cc n+1 2 & n is odd n 2 & n is even array . ) For ( n=1 ), we have ( f(1)= 1+1 2 =1 ) and, if ( n=2 ), we have ( f(2)= 2 2 =1 ) So, ( f(1)=f(2) ) but ( 1 2 ). Therefore, ( f(x) ) is not one-one. Now, ( f(x)= n+1 2 ) if ( n ) is odd if ( y= n+1 2 ), then ( n=2 y-1, y ) Also, ( f(x)= n 2 ) if ( n ) is even. That is, ( y= n 2 ) or ( n=2 y, y ) Therefore, ( f(x) ) is onto.

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