Surjectivity of $lceil x/2rceil$ over the integers [closed]

Clash Royale CLAN TAG#URR8PPP

up vote
1
down vote

favorite

Is the following function surjective from the set of integers to the set of integers?
$$lceil x/2rceil$$
My initial intuition says that it is, but I don't know if once the element $x$ from the domain starts getting higher in value (when $x$ approaches infinity), it would eventually miss an integer.

I hope this makes sense.

functions ceiling-function

share|cite|improve this question

edited Jul 23 at 15:10

Parcly Taxel

33.5k136588

asked Jul 23 at 15:08

Simon Garfe

104

closed as off-topic by amWhy, Xander Henderson, max_zorn, Isaac Browne, Parcly Taxel Jul 24 at 5:33

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Xander Henderson, max_zorn, Isaac Browne, Parcly Taxel

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  What happens if $x=2n,$ $n$ an integer?
  – Chris Leary
  Jul 23 at 15:13
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ChrisLeary you get the double value of n.
  – Simon Garfe
  Jul 23 at 15:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I believe you should get $n.$ Check Parclay's answer below.
  – Chris Leary
  Jul 23 at 16:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ChrisLeary yes, I understand now. If 2*n = x in the original equation.
  – Simon Garfe
  Jul 23 at 16:45
  

  
  
  
  

add a comment | 

up vote
1
down vote

favorite

Is the following function surjective from the set of integers to the set of integers?
$$lceil x/2rceil$$
My initial intuition says that it is, but I don't know if once the element $x$ from the domain starts getting higher in value (when $x$ approaches infinity), it would eventually miss an integer.

I hope this makes sense.

functions ceiling-function

share|cite|improve this question

edited Jul 23 at 15:10

Parcly Taxel

33.5k136588

asked Jul 23 at 15:08

Simon Garfe

104

closed as off-topic by amWhy, Xander Henderson, max_zorn, Isaac Browne, Parcly Taxel Jul 24 at 5:33

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Xander Henderson, max_zorn, Isaac Browne, Parcly Taxel

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  What happens if $x=2n,$ $n$ an integer?
  – Chris Leary
  Jul 23 at 15:13
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ChrisLeary you get the double value of n.
  – Simon Garfe
  Jul 23 at 15:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I believe you should get $n.$ Check Parclay's answer below.
  – Chris Leary
  Jul 23 at 16:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ChrisLeary yes, I understand now. If 2*n = x in the original equation.
  – Simon Garfe
  Jul 23 at 16:45
  

  
  
  
  

add a comment | 

up vote
1
down vote

favorite

up vote
1
down vote

favorite

Is the following function surjective from the set of integers to the set of integers?
$$lceil x/2rceil$$
My initial intuition says that it is, but I don't know if once the element $x$ from the domain starts getting higher in value (when $x$ approaches infinity), it would eventually miss an integer.

I hope this makes sense.

functions ceiling-function

share|cite|improve this question

edited Jul 23 at 15:10

Parcly Taxel

33.5k136588

asked Jul 23 at 15:08

Simon Garfe

104

Is the following function surjective from the set of integers to the set of integers?
$$lceil x/2rceil$$
My initial intuition says that it is, but I don't know if once the element $x$ from the domain starts getting higher in value (when $x$ approaches infinity), it would eventually miss an integer.

I hope this makes sense.

functions ceiling-function

share|cite|improve this question

edited Jul 23 at 15:10

Parcly Taxel

33.5k136588

asked Jul 23 at 15:08

Simon Garfe

104

share|cite|improve this question

share|cite|improve this question

edited Jul 23 at 15:10

Parcly Taxel

33.5k136588

edited Jul 23 at 15:10

Parcly Taxel

33.5k136588

edited Jul 23 at 15:10

Parcly Taxel

33.5k136588

33.5k136588

asked Jul 23 at 15:08

Simon Garfe

104

asked Jul 23 at 15:08

Simon Garfe

104

asked Jul 23 at 15:08

Simon Garfe

104

104

closed as off-topic by amWhy, Xander Henderson, max_zorn, Isaac Browne, Parcly Taxel Jul 24 at 5:33

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Xander Henderson, max_zorn, Isaac Browne, Parcly Taxel

If this question can be reworded to fit the rules in the help center, please edit the question.

closed as off-topic by amWhy, Xander Henderson, max_zorn, Isaac Browne, Parcly Taxel Jul 24 at 5:33

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, Xander Henderson, max_zorn, Isaac Browne, Parcly Taxel

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  What happens if $x=2n,$ $n$ an integer?
  – Chris Leary
  Jul 23 at 15:13
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ChrisLeary you get the double value of n.
  – Simon Garfe
  Jul 23 at 15:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I believe you should get $n.$ Check Parclay's answer below.
  – Chris Leary
  Jul 23 at 16:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ChrisLeary yes, I understand now. If 2*n = x in the original equation.
  – Simon Garfe
  Jul 23 at 16:45
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  What happens if $x=2n,$ $n$ an integer?
  – Chris Leary
  Jul 23 at 15:13
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ChrisLeary you get the double value of n.
  – Simon Garfe
  Jul 23 at 15:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I believe you should get $n.$ Check Parclay's answer below.
  – Chris Leary
  Jul 23 at 16:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @ChrisLeary yes, I understand now. If 2*n = x in the original equation.
  – Simon Garfe
  Jul 23 at 16:45
  

  
  
  
  

What happens if $x=2n,$ $n$ an integer?
– Chris Leary
Jul 23 at 15:13

What happens if $x=2n,$ $n$ an integer?
– Chris Leary
Jul 23 at 15:13

@ChrisLeary you get the double value of n.
– Simon Garfe
Jul 23 at 15:17

@ChrisLeary you get the double value of n.
– Simon Garfe
Jul 23 at 15:17

I believe you should get $n.$ Check Parclay's answer below.
– Chris Leary
Jul 23 at 16:35

I believe you should get $n.$ Check Parclay's answer below.
– Chris Leary
Jul 23 at 16:35

@ChrisLeary yes, I understand now. If 2*n = x in the original equation.
– Simon Garfe
Jul 23 at 16:45

@ChrisLeary yes, I understand now. If 2*n = x in the original equation.
– Simon Garfe
Jul 23 at 16:45

add a comment | 

2 Answers
2

active

oldest

votes

up vote
0
down vote



accepted

For any integer $y$ pick $x=2y$, which is an integer. Then $x/2$ is an integer, and since the ceiling of an integer is itself, $lceil x/2rceil=y$. Therefore the function has a preimage for every element in its codomain and is surjective.

share|cite|improve this answer

answered Jul 23 at 15:13

Parcly Taxel

33.5k136588

add a comment | 

up vote
1
down vote

Notice that $lceil (2n)/2rceil = n$.

share|cite|improve this answer

answered Jul 23 at 15:13

Fimpellizieri

16.4k11735

add a comment | 

2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
0
down vote



accepted

For any integer $y$ pick $x=2y$, which is an integer. Then $x/2$ is an integer, and since the ceiling of an integer is itself, $lceil x/2rceil=y$. Therefore the function has a preimage for every element in its codomain and is surjective.

share|cite|improve this answer

answered Jul 23 at 15:13

Parcly Taxel

33.5k136588

add a comment | 

up vote
0
down vote



accepted

For any integer $y$ pick $x=2y$, which is an integer. Then $x/2$ is an integer, and since the ceiling of an integer is itself, $lceil x/2rceil=y$. Therefore the function has a preimage for every element in its codomain and is surjective.

share|cite|improve this answer

answered Jul 23 at 15:13

Parcly Taxel

33.5k136588

add a comment | 

up vote
0
down vote



accepted

up vote
0
down vote



accepted

For any integer $y$ pick $x=2y$, which is an integer. Then $x/2$ is an integer, and since the ceiling of an integer is itself, $lceil x/2rceil=y$. Therefore the function has a preimage for every element in its codomain and is surjective.

share|cite|improve this answer

answered Jul 23 at 15:13

Parcly Taxel

33.5k136588

For any integer $y$ pick $x=2y$, which is an integer. Then $x/2$ is an integer, and since the ceiling of an integer is itself, $lceil x/2rceil=y$. Therefore the function has a preimage for every element in its codomain and is surjective.

share|cite|improve this answer

answered Jul 23 at 15:13

Parcly Taxel

33.5k136588

share|cite|improve this answer

share|cite|improve this answer

answered Jul 23 at 15:13

Parcly Taxel

33.5k136588

answered Jul 23 at 15:13

Parcly Taxel

33.5k136588

answered Jul 23 at 15:13

Parcly Taxel

33.5k136588

33.5k136588

add a comment | 

add a comment | 

up vote
1
down vote

Notice that $lceil (2n)/2rceil = n$.

share|cite|improve this answer

answered Jul 23 at 15:13

Fimpellizieri

16.4k11735

add a comment | 

up vote
1
down vote

Notice that $lceil (2n)/2rceil = n$.

share|cite|improve this answer

answered Jul 23 at 15:13

Fimpellizieri

16.4k11735

add a comment | 

up vote
1
down vote

up vote
1
down vote

Notice that $lceil (2n)/2rceil = n$.

share|cite|improve this answer

answered Jul 23 at 15:13

Fimpellizieri

16.4k11735

Notice that $lceil (2n)/2rceil = n$.

share|cite|improve this answer

answered Jul 23 at 15:13

Fimpellizieri

16.4k11735

share|cite|improve this answer

share|cite|improve this answer

answered Jul 23 at 15:13

Fimpellizieri

16.4k11735

answered Jul 23 at 15:13

Fimpellizieri

16.4k11735

answered Jul 23 at 15:13

Fimpellizieri

16.4k11735

16.4k11735

add a comment | 

add a comment |