Consider the functionf: Z→Z, n3+12n4Note: [] denotes the ceiling function.Which of the following is true?Of is not injective but is surjectiveOf is bijective.Of is neither injective nor surjective.Of is injective but is not surjective.

Answered: Consider the function f: Z→ Z, n→ T…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

Consider the function
f: Z→Z, n
3+12n
4
Note: [] denotes the ceiling function.
Which of the following is true?
Of is not injective but is surjective
Of is bijective.
Of is neither injective nor surjective.
Of is injective but is not surjective.

Expert Solution

Step 1: The Ceiling Function

The Ceiling Function is defined as the smallest integer that is not smaller than $x$ . It is the opposite of The Floor Function or the Greatest Integer function. It is denoted by $open ceil x close ceil$ .

For example $open ceil 4.5 close ceil space i s space 5$ whereas $open floor 4.5 close floor space i s space 4$ .

The given function is

$D e f i n e space a space f u n c t i o n space f colon space straight integer numbers rightwards arrow straight integer numbers space space b y space n rightwards arrow from bar open ceil fraction numerator 3 plus 12 n over denominator 4 end fraction close ceil f open parentheses n close parentheses equals open ceil fraction numerator 3 plus 12 n over denominator 4 end fraction close ceil f open parentheses n close parentheses equals open ceil 0.75 plus 3 n close ceil f open parentheses n close parentheses equals 3 n plus 1$

Check injective $f$ is one one function if $f open parentheses x close parentheses equals f open parentheses y close parentheses rightwards double arrow x equals y$

Check surjective $f colon space A rightwards arrow B$ is onto function if for every $y element of B space there exists space x element of A space s u c h space t h a t space f open parentheses x close parentheses equals y$