Prove function is injective, but not surjective Announcing the arrival of Valued Associate...
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Prove function is injective, but not surjective
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Prove that $f$ is NOT surjectiveInjective and Surjective Function ExamplesInjective/Surjective/Bijective questionSurjective and Not Injective ProblemProof writing involving functions: Injective and Surjectiveprove a function is injective/surjectiveinjective/surjective function that has ceiling and floorProve the following function is injective/surjectiveFind an infinite set $S$ and a function $g : S to S$ that is surjective but not injective.Is the given function injective, surjective?Determining whether the following is injective, surjective, bijective, or neither.
$begingroup$
Prove that the function $f(x) = x^2$ for $xin mathbb N$ is injective, but not surjective.
I know that injective means that $f(x)=f(y)implies x=y$, and
surjective means that for $f(x)=y$. So using this information, how would I prove this problem?
discrete-mathematics logic
edited Mar 26 at 16:41
Mauro ALLEGRANZA
68.3k449117
$endgroup$
|
show 2 more comments
$begingroup$
Prove that the function $f(x) = x^2$ for $xin mathbb N$ is injective, but not surjective.
I know that injective means that $f(x)=f(y)implies x=y$, and
surjective means that for $f(x)=y$. So using this information, how would I prove this problem?
discrete-mathematics logic
edited Mar 26 at 16:41
Mauro ALLEGRANZA
68.3k449117
$endgroup$
$begingroup$
Possible duplicate of Prove that $f$ is NOT surjective
$endgroup$
– Brian
Mar 26 at 1:49
2
$begingroup$
Injective: if $x^2=y^2$ then $(x+y)(x-y)=0$ so ...; Surjective: is there $xinmathbb N$ such that $x^2=2$?
$endgroup$
– J. W. Tanner
Mar 26 at 1:49
$begingroup$
How is x^2 not subjective?
$endgroup$
– user643202
Mar 26 at 1:51
1
$begingroup$
You have not specified a codomain.
$endgroup$
– N. F. Taussig
Mar 26 at 1:52
2
$begingroup$
As @N.F.Taussig noted, your definition of $f$ is technically incomplete. It is important to specify both what set $f$ 'takes inputs from' and what set $f$ 'produces elements from. If $f$ where given as a function $f:mathbb N to A$ where $A={ n^2 mid n in mathbb N }$, then $f$ would be surjective.
$endgroup$
– Brian
Mar 26 at 1:56
|
show 2 more comments
$begingroup$
Prove that the function $f(x) = x^2$ for $xin mathbb N$ is injective, but not surjective.
I know that injective means that $f(x)=f(y)implies x=y$, and
surjective means that for $f(x)=y$. So using this information, how would I prove this problem?
discrete-mathematics logic
edited Mar 26 at 16:41
Mauro ALLEGRANZA
68.3k449117
$endgroup$
Prove that the function $f(x) = x^2$ for $xin mathbb N$ is injective, but not surjective.
I know that injective means that $f(x)=f(y)implies x=y$, and
surjective means that for $f(x)=y$. So using this information, how would I prove this problem?
discrete-mathematics logic
discrete-mathematics logic
edited Mar 26 at 16:41
Mauro ALLEGRANZA
68.3k449117
edited Mar 26 at 16:41
Mauro ALLEGRANZA
68.3k449117
edited Mar 26 at 16:41
Mauro ALLEGRANZA
68.3k449117
edited Mar 26 at 16:41
Mauro ALLEGRANZA
68.3k449117
edited Mar 26 at 16:41
Mauro ALLEGRANZA
68.3k449117
68.3k449117
asked Mar 26 at 1:47
user643202
$begingroup$
Possible duplicate of Prove that $f$ is NOT surjective
$endgroup$
– Brian
Mar 26 at 1:49
2
$begingroup$
Injective: if $x^2=y^2$ then $(x+y)(x-y)=0$ so ...; Surjective: is there $xinmathbb N$ such that $x^2=2$?
$endgroup$
– J. W. Tanner
Mar 26 at 1:49
$begingroup$
How is x^2 not subjective?
$endgroup$
– user643202
Mar 26 at 1:51
1
$begingroup$
You have not specified a codomain.
$endgroup$
– N. F. Taussig
Mar 26 at 1:52
2
$begingroup$
As @N.F.Taussig noted, your definition of $f$ is technically incomplete. It is important to specify both what set $f$ 'takes inputs from' and what set $f$ 'produces elements from. If $f$ where given as a function $f:mathbb N to A$ where $A={ n^2 mid n in mathbb N }$, then $f$ would be surjective.
$endgroup$
– Brian
Mar 26 at 1:56
|
show 2 more comments
$begingroup$
Possible duplicate of Prove that $f$ is NOT surjective
$endgroup$
– Brian
Mar 26 at 1:49
2
$begingroup$
Injective: if $x^2=y^2$ then $(x+y)(x-y)=0$ so ...; Surjective: is there $xinmathbb N$ such that $x^2=2$?
$endgroup$
– J. W. Tanner
Mar 26 at 1:49
$begingroup$
How is x^2 not subjective?
$endgroup$
– user643202
Mar 26 at 1:51
1
$begingroup$
You have not specified a codomain.
$endgroup$
– N. F. Taussig
Mar 26 at 1:52
2
$begingroup$
As @N.F.Taussig noted, your definition of $f$ is technically incomplete. It is important to specify both what set $f$ 'takes inputs from' and what set $f$ 'produces elements from. If $f$ where given as a function $f:mathbb N to A$ where $A={ n^2 mid n in mathbb N }$, then $f$ would be surjective.
$endgroup$
– Brian
Mar 26 at 1:56
$begingroup$
Possible duplicate of Prove that $f$ is NOT surjective
$endgroup$
– Brian
Mar 26 at 1:49
$begingroup$
Possible duplicate of Prove that $f$ is NOT surjective
$endgroup$
– Brian
Mar 26 at 1:49
2
2
$begingroup$
Injective: if $x^2=y^2$ then $(x+y)(x-y)=0$ so ...; Surjective: is there $xinmathbb N$ such that $x^2=2$?
$endgroup$
– J. W. Tanner
Mar 26 at 1:49
$begingroup$
Injective: if $x^2=y^2$ then $(x+y)(x-y)=0$ so ...; Surjective: is there $xinmathbb N$ such that $x^2=2$?
$endgroup$
– J. W. Tanner
Mar 26 at 1:49
$begingroup$
How is x^2 not subjective?
$endgroup$
– user643202
Mar 26 at 1:51
$begingroup$
How is x^2 not subjective?
$endgroup$
– user643202
Mar 26 at 1:51
1
1
$begingroup$
You have not specified a codomain.
$endgroup$
– N. F. Taussig
Mar 26 at 1:52
$begingroup$
You have not specified a codomain.
$endgroup$
– N. F. Taussig
Mar 26 at 1:52
2
2
$begingroup$
As @N.F.Taussig noted, your definition of $f$ is technically incomplete. It is important to specify both what set $f$ 'takes inputs from' and what set $f$ 'produces elements from. If $f$ where given as a function $f:mathbb N to A$ where $A={ n^2 mid n in mathbb N }$, then $f$ would be surjective.
$endgroup$
– Brian
Mar 26 at 1:56
$begingroup$
As @N.F.Taussig noted, your definition of $f$ is technically incomplete. It is important to specify both what set $f$ 'takes inputs from' and what set $f$ 'produces elements from. If $f$ where given as a function $f:mathbb N to A$ where $A={ n^2 mid n in mathbb N }$, then $f$ would be surjective.
$endgroup$
– Brian
Mar 26 at 1:56
|
show 2 more comments
2 Answers
2
active
oldest
votes
$begingroup$
It seems that you mean $f: mathbb{N} to mathbb{N}$ given by $x mapsto x^2$ is injective and not surjective.
To show it is not surjective, hint:
Can you find a natural number that isn't a square of some natural number?
To show it is injective, if $x^2 = y^2$ we have $x = y$ or $x = -y$, can you continue from here?
edited Mar 26 at 2:13
J. W. Tanner
5,0751520
answered Mar 26 at 2:02
MariahMariah
2,1471718
$endgroup$
$begingroup$
So like numbers like 3, 5, and 7?
$endgroup$
– user643202
Mar 26 at 2:03
$begingroup$
@Anonymous those are examples yes. Do you understand why that shows the function isn't a surjection?
$endgroup$
– Mariah
Mar 26 at 2:04
$begingroup$
Wouldn't x = -y not make it injective?
$endgroup$
– user643202
Mar 26 at 2:05
$begingroup$
@Anonymous what is our domain?
$endgroup$
– Mariah
Mar 26 at 2:06
$begingroup$
all natural numbers
$endgroup$
– user643202
Mar 26 at 2:07
|
show 1 more comment
$begingroup$
You just need to give a counter-example here. Consider the number $y=17 in mathbb{N}$. Then $x^2=y Rightarrow x^2 = 17 Rightarrow x = pmsqrt{17} notin mathbb{N}.$ You have found a $y$ such that $x notin mathbb{N}$ for $f(x)=y$. Therefore, $f$ is not surjective.
answered Mar 26 at 7:14
YFPYFP
3912
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
It seems that you mean $f: mathbb{N} to mathbb{N}$ given by $x mapsto x^2$ is injective and not surjective.
To show it is not surjective, hint:
Can you find a natural number that isn't a square of some natural number?
To show it is injective, if $x^2 = y^2$ we have $x = y$ or $x = -y$, can you continue from here?
edited Mar 26 at 2:13
J. W. Tanner
5,0751520
answered Mar 26 at 2:02
MariahMariah
2,1471718
$endgroup$
$begingroup$
So like numbers like 3, 5, and 7?
$endgroup$
– user643202
Mar 26 at 2:03
$begingroup$
@Anonymous those are examples yes. Do you understand why that shows the function isn't a surjection?
$endgroup$
– Mariah
Mar 26 at 2:04
$begingroup$
Wouldn't x = -y not make it injective?
$endgroup$
– user643202
Mar 26 at 2:05
$begingroup$
@Anonymous what is our domain?
$endgroup$
– Mariah
Mar 26 at 2:06
$begingroup$
all natural numbers
$endgroup$
– user643202
Mar 26 at 2:07
|
show 1 more comment
$begingroup$
It seems that you mean $f: mathbb{N} to mathbb{N}$ given by $x mapsto x^2$ is injective and not surjective.
To show it is not surjective, hint:
Can you find a natural number that isn't a square of some natural number?
To show it is injective, if $x^2 = y^2$ we have $x = y$ or $x = -y$, can you continue from here?
edited Mar 26 at 2:13
J. W. Tanner
5,0751520
answered Mar 26 at 2:02
MariahMariah
2,1471718
$endgroup$
$begingroup$
So like numbers like 3, 5, and 7?
$endgroup$
– user643202
Mar 26 at 2:03
$begingroup$
@Anonymous those are examples yes. Do you understand why that shows the function isn't a surjection?
$endgroup$
– Mariah
Mar 26 at 2:04
$begingroup$
Wouldn't x = -y not make it injective?
$endgroup$
– user643202
Mar 26 at 2:05
$begingroup$
@Anonymous what is our domain?
$endgroup$
– Mariah
Mar 26 at 2:06
$begingroup$
all natural numbers
$endgroup$
– user643202
Mar 26 at 2:07
|
show 1 more comment
$begingroup$
It seems that you mean $f: mathbb{N} to mathbb{N}$ given by $x mapsto x^2$ is injective and not surjective.
To show it is not surjective, hint:
Can you find a natural number that isn't a square of some natural number?
To show it is injective, if $x^2 = y^2$ we have $x = y$ or $x = -y$, can you continue from here?
edited Mar 26 at 2:13
J. W. Tanner
5,0751520
answered Mar 26 at 2:02
MariahMariah
2,1471718
$endgroup$
It seems that you mean $f: mathbb{N} to mathbb{N}$ given by $x mapsto x^2$ is injective and not surjective.
To show it is not surjective, hint:
Can you find a natural number that isn't a square of some natural number?
To show it is injective, if $x^2 = y^2$ we have $x = y$ or $x = -y$, can you continue from here?
edited Mar 26 at 2:13
J. W. Tanner
5,0751520
answered Mar 26 at 2:02
MariahMariah
2,1471718
edited Mar 26 at 2:13
J. W. Tanner
5,0751520
edited Mar 26 at 2:13
J. W. Tanner
5,0751520
edited Mar 26 at 2:13
J. W. Tanner
5,0751520
5,0751520
answered Mar 26 at 2:02
MariahMariah
2,1471718
answered Mar 26 at 2:02
MariahMariah
2,1471718
answered Mar 26 at 2:02
MariahMariah
2,1471718
2,1471718
$begingroup$
So like numbers like 3, 5, and 7?
$endgroup$
– user643202
Mar 26 at 2:03
$begingroup$
@Anonymous those are examples yes. Do you understand why that shows the function isn't a surjection?
$endgroup$
– Mariah
Mar 26 at 2:04
$begingroup$
Wouldn't x = -y not make it injective?
$endgroup$
– user643202
Mar 26 at 2:05
$begingroup$
@Anonymous what is our domain?
$endgroup$
– Mariah
Mar 26 at 2:06
$begingroup$
all natural numbers
$endgroup$
– user643202
Mar 26 at 2:07
|
show 1 more comment
$begingroup$
So like numbers like 3, 5, and 7?
$endgroup$
– user643202
Mar 26 at 2:03
$begingroup$
@Anonymous those are examples yes. Do you understand why that shows the function isn't a surjection?
$endgroup$
– Mariah
Mar 26 at 2:04
$begingroup$
Wouldn't x = -y not make it injective?
$endgroup$
– user643202
Mar 26 at 2:05
$begingroup$
@Anonymous what is our domain?
$endgroup$
– Mariah
Mar 26 at 2:06
$begingroup$
all natural numbers
$endgroup$
– user643202
Mar 26 at 2:07
$begingroup$
So like numbers like 3, 5, and 7?
$endgroup$
– user643202
Mar 26 at 2:03
$begingroup$
So like numbers like 3, 5, and 7?
$endgroup$
– user643202
Mar 26 at 2:03
$begingroup$
@Anonymous those are examples yes. Do you understand why that shows the function isn't a surjection?
$endgroup$
– Mariah
Mar 26 at 2:04
$begingroup$
@Anonymous those are examples yes. Do you understand why that shows the function isn't a surjection?
$endgroup$
– Mariah
Mar 26 at 2:04
$begingroup$
Wouldn't x = -y not make it injective?
$endgroup$
– user643202
Mar 26 at 2:05
$begingroup$
Wouldn't x = -y not make it injective?
$endgroup$
– user643202
Mar 26 at 2:05
$begingroup$
@Anonymous what is our domain?
$endgroup$
– Mariah
Mar 26 at 2:06
$begingroup$
@Anonymous what is our domain?
$endgroup$
– Mariah
Mar 26 at 2:06
$begingroup$
all natural numbers
$endgroup$
– user643202
Mar 26 at 2:07
$begingroup$
all natural numbers
$endgroup$
– user643202
Mar 26 at 2:07
|
show 1 more comment
$begingroup$
You just need to give a counter-example here. Consider the number $y=17 in mathbb{N}$. Then $x^2=y Rightarrow x^2 = 17 Rightarrow x = pmsqrt{17} notin mathbb{N}.$ You have found a $y$ such that $x notin mathbb{N}$ for $f(x)=y$. Therefore, $f$ is not surjective.
answered Mar 26 at 7:14
YFPYFP
3912
$endgroup$
add a comment |
$begingroup$
You just need to give a counter-example here. Consider the number $y=17 in mathbb{N}$. Then $x^2=y Rightarrow x^2 = 17 Rightarrow x = pmsqrt{17} notin mathbb{N}.$ You have found a $y$ such that $x notin mathbb{N}$ for $f(x)=y$. Therefore, $f$ is not surjective.
answered Mar 26 at 7:14
YFPYFP
3912
$endgroup$
add a comment |
$begingroup$
You just need to give a counter-example here. Consider the number $y=17 in mathbb{N}$. Then $x^2=y Rightarrow x^2 = 17 Rightarrow x = pmsqrt{17} notin mathbb{N}.$ You have found a $y$ such that $x notin mathbb{N}$ for $f(x)=y$. Therefore, $f$ is not surjective.
answered Mar 26 at 7:14
YFPYFP
3912
$endgroup$
You just need to give a counter-example here. Consider the number $y=17 in mathbb{N}$. Then $x^2=y Rightarrow x^2 = 17 Rightarrow x = pmsqrt{17} notin mathbb{N}.$ You have found a $y$ such that $x notin mathbb{N}$ for $f(x)=y$. Therefore, $f$ is not surjective.
answered Mar 26 at 7:14
YFPYFP
3912
answered Mar 26 at 7:14
YFPYFP
3912
answered Mar 26 at 7:14
YFPYFP
3912
answered Mar 26 at 7:14
YFPYFP
3912
3912
add a comment |
add a comment |
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$begingroup$
Possible duplicate of Prove that $f$ is NOT surjective
$endgroup$
– Brian
Mar 26 at 1:49
2
$begingroup$
Injective: if $x^2=y^2$ then $(x+y)(x-y)=0$ so ...; Surjective: is there $xinmathbb N$ such that $x^2=2$?
$endgroup$
– J. W. Tanner
Mar 26 at 1:49
$begingroup$
How is x^2 not subjective?
$endgroup$
– user643202
Mar 26 at 1:51
1
$begingroup$
You have not specified a codomain.
$endgroup$
– N. F. Taussig
Mar 26 at 1:52
2
$begingroup$
As @N.F.Taussig noted, your definition of $f$ is technically incomplete. It is important to specify both what set $f$ 'takes inputs from' and what set $f$ 'produces elements from. If $f$ where given as a function $f:mathbb N to A$ where $A={ n^2 mid n in mathbb N }$, then $f$ would be surjective.
$endgroup$
– Brian
Mar 26 at 1:56