Help proving that $,fcirc g$ and $,hcirc f$ are injective if and only if $,g,f,h$ are bijective [closed]
$begingroup$
Let $g,f,h;$ be functions and $A,B,C,$ subsets of $mathbb{R};$such that $g:Arightarrow B,; f:Brightarrow C;text{and}; h:Crightarrow D$. Prove the following:
$$fcirc g,;hcirc f;;;text{injective} Leftarrow Rightarrow g,f,h;;;text{bijective}$$
L.H.S:
If $g$ and $f$ are bijective functions, then so it is their composition and in particular injective. And the same with $h$ and $f$.
When proving it R.H.S, I have problems because I don't know how should I approach the demonstration.
If $fcirc g$ is injective, then both $f$ and $g$ are injective, but now how do I prove that they also have to be surjective?
functions elementary-set-theory
asked Jan 9 at 0:01
RoberRober
11
$endgroup$
closed as off-topic by Eevee Trainer, Leucippus, Shailesh, mrtaurho, Lord Shark the Unknown Jan 9 at 7:09
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 provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Eevee Trainer, Leucippus, Shailesh, mrtaurho, Lord Shark the Unknown
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Let $g,f,h;$ be functions and $A,B,C,$ subsets of $mathbb{R};$such that $g:Arightarrow B,; f:Brightarrow C;text{and}; h:Crightarrow D$. Prove the following:
$$fcirc g,;hcirc f;;;text{injective} Leftarrow Rightarrow g,f,h;;;text{bijective}$$
L.H.S:
If $g$ and $f$ are bijective functions, then so it is their composition and in particular injective. And the same with $h$ and $f$.
When proving it R.H.S, I have problems because I don't know how should I approach the demonstration.
If $fcirc g$ is injective, then both $f$ and $g$ are injective, but now how do I prove that they also have to be surjective?
functions elementary-set-theory
asked Jan 9 at 0:01
RoberRober
11
$endgroup$
closed as off-topic by Eevee Trainer, Leucippus, Shailesh, mrtaurho, Lord Shark the Unknown Jan 9 at 7:09
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 provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Eevee Trainer, Leucippus, Shailesh, mrtaurho, Lord Shark the Unknown
If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
Something is wrong here: $g circ f$ doesn't make any sense. since the image of $f$ is contained in $C$, but the domain of $g$ is $A$. Do you perhaps mean $f circ g$? Either way, the result is still false: take $g$ and $h$ to be any bijection, and $f$ to be any non-surjective injection, and clearly $fcirc g$ and $f circ h$ are injective, but $f$ is not surjective.
$endgroup$
– user3482749
Jan 9 at 0:04
$begingroup$
$g circ f$ is not well defined. You probably mean to consider $f circ g$ and $hcirc f$.
$endgroup$
– Snake707
Jan 9 at 0:05
$begingroup$
Yes, I meant $fcirc g,$ and $hcirc f$.
$endgroup$
– Rober
Jan 9 at 0:12
add a comment |
$begingroup$
Let $g,f,h;$ be functions and $A,B,C,$ subsets of $mathbb{R};$such that $g:Arightarrow B,; f:Brightarrow C;text{and}; h:Crightarrow D$. Prove the following:
$$fcirc g,;hcirc f;;;text{injective} Leftarrow Rightarrow g,f,h;;;text{bijective}$$
L.H.S:
If $g$ and $f$ are bijective functions, then so it is their composition and in particular injective. And the same with $h$ and $f$.
When proving it R.H.S, I have problems because I don't know how should I approach the demonstration.
If $fcirc g$ is injective, then both $f$ and $g$ are injective, but now how do I prove that they also have to be surjective?
functions elementary-set-theory
asked Jan 9 at 0:01
RoberRober
11
$endgroup$
Let $g,f,h;$ be functions and $A,B,C,$ subsets of $mathbb{R};$such that $g:Arightarrow B,; f:Brightarrow C;text{and}; h:Crightarrow D$. Prove the following:
$$fcirc g,;hcirc f;;;text{injective} Leftarrow Rightarrow g,f,h;;;text{bijective}$$
L.H.S:
If $g$ and $f$ are bijective functions, then so it is their composition and in particular injective. And the same with $h$ and $f$.
When proving it R.H.S, I have problems because I don't know how should I approach the demonstration.
If $fcirc g$ is injective, then both $f$ and $g$ are injective, but now how do I prove that they also have to be surjective?
functions elementary-set-theory
functions elementary-set-theory
asked Jan 9 at 0:01
RoberRober
11
asked Jan 9 at 0:01
RoberRober
11
edited Jan 11 at 12:19
Rober
asked Jan 9 at 0:01
RoberRober
11
asked Jan 9 at 0:01
RoberRober
11
asked Jan 9 at 0:01
RoberRober
11
11
closed as off-topic by Eevee Trainer, Leucippus, Shailesh, mrtaurho, Lord Shark the Unknown Jan 9 at 7:09
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 provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Eevee Trainer, Leucippus, Shailesh, mrtaurho, Lord Shark the Unknown
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Eevee Trainer, Leucippus, Shailesh, mrtaurho, Lord Shark the Unknown Jan 9 at 7:09
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 provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Eevee Trainer, Leucippus, Shailesh, mrtaurho, Lord Shark the Unknown
If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
Something is wrong here: $g circ f$ doesn't make any sense. since the image of $f$ is contained in $C$, but the domain of $g$ is $A$. Do you perhaps mean $f circ g$? Either way, the result is still false: take $g$ and $h$ to be any bijection, and $f$ to be any non-surjective injection, and clearly $fcirc g$ and $f circ h$ are injective, but $f$ is not surjective.
$endgroup$
– user3482749
Jan 9 at 0:04
$begingroup$
$g circ f$ is not well defined. You probably mean to consider $f circ g$ and $hcirc f$.
$endgroup$
– Snake707
Jan 9 at 0:05
$begingroup$
Yes, I meant $fcirc g,$ and $hcirc f$.
$endgroup$
– Rober
Jan 9 at 0:12
add a comment |
2
$begingroup$
Something is wrong here: $g circ f$ doesn't make any sense. since the image of $f$ is contained in $C$, but the domain of $g$ is $A$. Do you perhaps mean $f circ g$? Either way, the result is still false: take $g$ and $h$ to be any bijection, and $f$ to be any non-surjective injection, and clearly $fcirc g$ and $f circ h$ are injective, but $f$ is not surjective.
$endgroup$
– user3482749
Jan 9 at 0:04
$begingroup$
$g circ f$ is not well defined. You probably mean to consider $f circ g$ and $hcirc f$.
$endgroup$
– Snake707
Jan 9 at 0:05
$begingroup$
Yes, I meant $fcirc g,$ and $hcirc f$.
$endgroup$
– Rober
Jan 9 at 0:12
2
2
$begingroup$
Something is wrong here: $g circ f$ doesn't make any sense. since the image of $f$ is contained in $C$, but the domain of $g$ is $A$. Do you perhaps mean $f circ g$? Either way, the result is still false: take $g$ and $h$ to be any bijection, and $f$ to be any non-surjective injection, and clearly $fcirc g$ and $f circ h$ are injective, but $f$ is not surjective.
$endgroup$
– user3482749
Jan 9 at 0:04
$begingroup$
Something is wrong here: $g circ f$ doesn't make any sense. since the image of $f$ is contained in $C$, but the domain of $g$ is $A$. Do you perhaps mean $f circ g$? Either way, the result is still false: take $g$ and $h$ to be any bijection, and $f$ to be any non-surjective injection, and clearly $fcirc g$ and $f circ h$ are injective, but $f$ is not surjective.
$endgroup$
– user3482749
Jan 9 at 0:04
$begingroup$
$g circ f$ is not well defined. You probably mean to consider $f circ g$ and $hcirc f$.
$endgroup$
– Snake707
Jan 9 at 0:05
$begingroup$
$g circ f$ is not well defined. You probably mean to consider $f circ g$ and $hcirc f$.
$endgroup$
– Snake707
Jan 9 at 0:05
$begingroup$
Yes, I meant $fcirc g,$ and $hcirc f$.
$endgroup$
– Rober
Jan 9 at 0:12
$begingroup$
Yes, I meant $fcirc g,$ and $hcirc f$.
$endgroup$
– Rober
Jan 9 at 0:12
add a comment |
0
active
oldest
votes
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
2
$begingroup$
Something is wrong here: $g circ f$ doesn't make any sense. since the image of $f$ is contained in $C$, but the domain of $g$ is $A$. Do you perhaps mean $f circ g$? Either way, the result is still false: take $g$ and $h$ to be any bijection, and $f$ to be any non-surjective injection, and clearly $fcirc g$ and $f circ h$ are injective, but $f$ is not surjective.
$endgroup$
– user3482749
Jan 9 at 0:04
$begingroup$
$g circ f$ is not well defined. You probably mean to consider $f circ g$ and $hcirc f$.
$endgroup$
– Snake707
Jan 9 at 0:05
$begingroup$
Yes, I meant $fcirc g,$ and $hcirc f$.
$endgroup$
– Rober
Jan 9 at 0:12