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L e t space f colon N rightwards arrow N space b e space d e f i n e d space b y space
f open parentheses n close parentheses equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator n plus 1 over denominator 2 end fraction comma space i f space n space i s space o d d end cell row cell n over 2 comma space i f space n space i s space e v e n end cell end table close comma space f o r space a l l space n element of N
S t a t e space w h e t h e r space t h e space f u n c t i o n space f space i s space b i j e c t i v e. space J u s t i f y space y o u r space a n s w e r.

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Asked by Topperlearning User 25th October 2016, 8:59 AM
Answered by Expert
Answer:

G i v e n space f colon N rightwards arrow N space b e space d e f i n e d space b y space
f open parentheses n close parentheses equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator n plus 1 over denominator 2 end fraction comma space i f space n space i s space o d d end cell row cell n over 2 comma space i f space n space i s space e v e n end cell end table close comma space f o r space a l l space n element of N
W e space n o t e space t h a t space
f open parentheses 1 close parentheses equals fraction numerator 1 plus 1 over denominator 2 end fraction equals 1 comma space f open parentheses 2 close parentheses equals 2 over 2 equals 1
f open parentheses 3 close parentheses equals fraction numerator 3 plus 1 over denominator 2 end fraction equals 2 comma space f open parentheses 4 close parentheses equals 4 over 2 equals 2
I n space g e n e r a l comma
f open parentheses 2 m minus 1 close parentheses equals fraction numerator open parentheses 2 m minus 1 close parentheses plus 1 over denominator 2 end fraction equals m space a n d space f open parentheses 2 m close parentheses equals fraction numerator 2 m over denominator 2 end fraction equals m
rightwards double arrow f open parentheses 2 m minus 1 close parentheses equals f open parentheses 2 m close parentheses comma space w h e r e space m space i s space a n y space p o s i t i v e space i n t e g e r.
rightwards double arrow f space i s space n o t space o n e minus o n e
F o r space a n y space n element of N comma space t h e r e space i s space 2 n element of N comma space s u c h space t h a t space f open parentheses 2 n close parentheses equals fraction numerator 2 n over denominator 2 end fraction equals n
T h u s comma space f space i s space o n t o space b u t space n o t space o n e minus o n e. space
T h e r e f o r e comma space f space i s space n o t space a space b i j e c t i o n.

Answered by Expert 25th October 2016, 10:59 AM
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