Is it true that $Asubset f^{-1}(f(A))$ if and only if $f$ is injective?
I'm doing some homework and I have trouble with some problems.
Let $X,Y$ be sets, $f:Xmapsto Y$, $A subset X$, $Csubset Y$.
I need to prove that $Asubset f^{-1}(f(A))$ if and only if $f$ is injective; and that $f(f^{-1}(C))subset C$ if and only if $f$ is surjective.
I tried to apply the definitions but I just got confused because of $f^{-1}(f(A))$, so i could use some advice.
analysis elementary-set-theory
asked Nov 26 at 22:14
Armando Rosas
525
add a comment |
I'm doing some homework and I have trouble with some problems.
Let $X,Y$ be sets, $f:Xmapsto Y$, $A subset X$, $Csubset Y$.
I need to prove that $Asubset f^{-1}(f(A))$ if and only if $f$ is injective; and that $f(f^{-1}(C))subset C$ if and only if $f$ is surjective.
I tried to apply the definitions but I just got confused because of $f^{-1}(f(A))$, so i could use some advice.
analysis elementary-set-theory
asked Nov 26 at 22:14
Armando Rosas
525
It is not enough that $A subset f^{-1}(f(A))$ for some $A subset X$, because let $f: {1, 2, 3} rightarrow {1, 2}$ with $f(1) = 1$, $f(2) = f(3) = 2$ and $A = {1}$ then $A subset f^{-1}(f(A))$ but f is not injective.
– BurningKarl
Nov 26 at 22:24
These properties are not true as stated. Are you sure this is what your homework says?
– Arnaud D.
Nov 26 at 22:25
@ArnaudD I'm not sure. That's how I have the exercises in the copy my teacher gave me.
– Armando Rosas
Nov 27 at 0:26
add a comment |
I'm doing some homework and I have trouble with some problems.
Let $X,Y$ be sets, $f:Xmapsto Y$, $A subset X$, $Csubset Y$.
I need to prove that $Asubset f^{-1}(f(A))$ if and only if $f$ is injective; and that $f(f^{-1}(C))subset C$ if and only if $f$ is surjective.
I tried to apply the definitions but I just got confused because of $f^{-1}(f(A))$, so i could use some advice.
analysis elementary-set-theory
asked Nov 26 at 22:14
Armando Rosas
525
I'm doing some homework and I have trouble with some problems.
Let $X,Y$ be sets, $f:Xmapsto Y$, $A subset X$, $Csubset Y$.
I need to prove that $Asubset f^{-1}(f(A))$ if and only if $f$ is injective; and that $f(f^{-1}(C))subset C$ if and only if $f$ is surjective.
I tried to apply the definitions but I just got confused because of $f^{-1}(f(A))$, so i could use some advice.
analysis elementary-set-theory
analysis elementary-set-theory
asked Nov 26 at 22:14
Armando Rosas
525
asked Nov 26 at 22:14
Armando Rosas
525
edited Nov 27 at 0:20
asked Nov 26 at 22:14
Armando Rosas
525
asked Nov 26 at 22:14
Armando Rosas
525
asked Nov 26 at 22:14
Armando Rosas
525
525
It is not enough that $A subset f^{-1}(f(A))$ for some $A subset X$, because let $f: {1, 2, 3} rightarrow {1, 2}$ with $f(1) = 1$, $f(2) = f(3) = 2$ and $A = {1}$ then $A subset f^{-1}(f(A))$ but f is not injective.
– BurningKarl
Nov 26 at 22:24
These properties are not true as stated. Are you sure this is what your homework says?
– Arnaud D.
Nov 26 at 22:25
@ArnaudD I'm not sure. That's how I have the exercises in the copy my teacher gave me.
– Armando Rosas
Nov 27 at 0:26
add a comment |
It is not enough that $A subset f^{-1}(f(A))$ for some $A subset X$, because let $f: {1, 2, 3} rightarrow {1, 2}$ with $f(1) = 1$, $f(2) = f(3) = 2$ and $A = {1}$ then $A subset f^{-1}(f(A))$ but f is not injective.
– BurningKarl
Nov 26 at 22:24
These properties are not true as stated. Are you sure this is what your homework says?
– Arnaud D.
Nov 26 at 22:25
@ArnaudD I'm not sure. That's how I have the exercises in the copy my teacher gave me.
– Armando Rosas
Nov 27 at 0:26
It is not enough that $A subset f^{-1}(f(A))$ for some $A subset X$, because let $f: {1, 2, 3} rightarrow {1, 2}$ with $f(1) = 1$, $f(2) = f(3) = 2$ and $A = {1}$ then $A subset f^{-1}(f(A))$ but f is not injective.
– BurningKarl
Nov 26 at 22:24
It is not enough that $A subset f^{-1}(f(A))$ for some $A subset X$, because let $f: {1, 2, 3} rightarrow {1, 2}$ with $f(1) = 1$, $f(2) = f(3) = 2$ and $A = {1}$ then $A subset f^{-1}(f(A))$ but f is not injective.
– BurningKarl
Nov 26 at 22:24
These properties are not true as stated. Are you sure this is what your homework says?
– Arnaud D.
Nov 26 at 22:25
These properties are not true as stated. Are you sure this is what your homework says?
– Arnaud D.
Nov 26 at 22:25
@ArnaudD I'm not sure. That's how I have the exercises in the copy my teacher gave me.
– Armando Rosas
Nov 27 at 0:26
@ArnaudD I'm not sure. That's how I have the exercises in the copy my teacher gave me.
– Armando Rosas
Nov 27 at 0:26
add a comment |
1 Answer
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Note that $f^{-1}(f(C))$ should be $f(f^{-1}(C))$.
The inclusion $Asubset f^{-1}(f(A))$ is true for every map and every subset of the map's domain.
Indeed, if $ain A$, then $f(a)in f(A)$ (by definition), which is to say that $ain f^{-1}(f(A))$.
Similarly, $f(f^{-1}(C))subset C$ holds for every map $f$ and every subset of the map's codomain. Indeed, if $yin f(f^{-1}(C))$, then $y=f(x)$, for some $xin f^{-1}(C)$. But then $f(x)in C$, which is to say $yin C$.
What you want to show is that
$fcolon Xto Y$ is injective if and only if, for every $Asubset X$, $A=f^{-1}(f(A))$
and similarly that
$fcolon Xto Y$ is surjective if and only if, for every $Csubset Y$, $C=f(f^{-1}(C))$
Hint for the $Rightarrow$ direction in the first statement: you have to prove that $f^{-1}(f(A))subset A$. Take $xin f^{-1}(f(A))$; then $f(x)in f(A)$. Apply injectivity.
answered Nov 26 at 22:24
egreg
178k1484200
I made the correction. The same hint holds for the surjectivity proposition?
– Armando Rosas
Nov 27 at 0:24
@ArmandoRosas Very similar. Don't forget to prove also the $Leftarrow$ directions.
– egreg
Nov 27 at 7:46
I have trouble with the $Leftarrow$ proof of surjectivity. I let c be in C , then c it's an element of $f(f^{-1}(C))$, can i directly say that f is surjective here? Because then $c=f(x)$ with $xin f^{-1}(C)subset X$, is it not?
– Armando Rosas
Nov 27 at 22:03
@ArmandoRosas For $C=Y$, $Y=f(f^{-1}(Y))=f(X)$.
– egreg
Nov 27 at 22:19
add a comment |
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Note that $f^{-1}(f(C))$ should be $f(f^{-1}(C))$.
The inclusion $Asubset f^{-1}(f(A))$ is true for every map and every subset of the map's domain.
Indeed, if $ain A$, then $f(a)in f(A)$ (by definition), which is to say that $ain f^{-1}(f(A))$.
Similarly, $f(f^{-1}(C))subset C$ holds for every map $f$ and every subset of the map's codomain. Indeed, if $yin f(f^{-1}(C))$, then $y=f(x)$, for some $xin f^{-1}(C)$. But then $f(x)in C$, which is to say $yin C$.
What you want to show is that
$fcolon Xto Y$ is injective if and only if, for every $Asubset X$, $A=f^{-1}(f(A))$
and similarly that
$fcolon Xto Y$ is surjective if and only if, for every $Csubset Y$, $C=f(f^{-1}(C))$
Hint for the $Rightarrow$ direction in the first statement: you have to prove that $f^{-1}(f(A))subset A$. Take $xin f^{-1}(f(A))$; then $f(x)in f(A)$. Apply injectivity.
answered Nov 26 at 22:24
egreg
178k1484200
I made the correction. The same hint holds for the surjectivity proposition?
– Armando Rosas
Nov 27 at 0:24
@ArmandoRosas Very similar. Don't forget to prove also the $Leftarrow$ directions.
– egreg
Nov 27 at 7:46
I have trouble with the $Leftarrow$ proof of surjectivity. I let c be in C , then c it's an element of $f(f^{-1}(C))$, can i directly say that f is surjective here? Because then $c=f(x)$ with $xin f^{-1}(C)subset X$, is it not?
– Armando Rosas
Nov 27 at 22:03
@ArmandoRosas For $C=Y$, $Y=f(f^{-1}(Y))=f(X)$.
– egreg
Nov 27 at 22:19
add a comment |
Note that $f^{-1}(f(C))$ should be $f(f^{-1}(C))$.
The inclusion $Asubset f^{-1}(f(A))$ is true for every map and every subset of the map's domain.
Indeed, if $ain A$, then $f(a)in f(A)$ (by definition), which is to say that $ain f^{-1}(f(A))$.
Similarly, $f(f^{-1}(C))subset C$ holds for every map $f$ and every subset of the map's codomain. Indeed, if $yin f(f^{-1}(C))$, then $y=f(x)$, for some $xin f^{-1}(C)$. But then $f(x)in C$, which is to say $yin C$.
What you want to show is that
$fcolon Xto Y$ is injective if and only if, for every $Asubset X$, $A=f^{-1}(f(A))$
and similarly that
$fcolon Xto Y$ is surjective if and only if, for every $Csubset Y$, $C=f(f^{-1}(C))$
Hint for the $Rightarrow$ direction in the first statement: you have to prove that $f^{-1}(f(A))subset A$. Take $xin f^{-1}(f(A))$; then $f(x)in f(A)$. Apply injectivity.
answered Nov 26 at 22:24
egreg
178k1484200
I made the correction. The same hint holds for the surjectivity proposition?
– Armando Rosas
Nov 27 at 0:24
@ArmandoRosas Very similar. Don't forget to prove also the $Leftarrow$ directions.
– egreg
Nov 27 at 7:46
I have trouble with the $Leftarrow$ proof of surjectivity. I let c be in C , then c it's an element of $f(f^{-1}(C))$, can i directly say that f is surjective here? Because then $c=f(x)$ with $xin f^{-1}(C)subset X$, is it not?
– Armando Rosas
Nov 27 at 22:03
@ArmandoRosas For $C=Y$, $Y=f(f^{-1}(Y))=f(X)$.
– egreg
Nov 27 at 22:19
add a comment |
Note that $f^{-1}(f(C))$ should be $f(f^{-1}(C))$.
The inclusion $Asubset f^{-1}(f(A))$ is true for every map and every subset of the map's domain.
Indeed, if $ain A$, then $f(a)in f(A)$ (by definition), which is to say that $ain f^{-1}(f(A))$.
Similarly, $f(f^{-1}(C))subset C$ holds for every map $f$ and every subset of the map's codomain. Indeed, if $yin f(f^{-1}(C))$, then $y=f(x)$, for some $xin f^{-1}(C)$. But then $f(x)in C$, which is to say $yin C$.
What you want to show is that
$fcolon Xto Y$ is injective if and only if, for every $Asubset X$, $A=f^{-1}(f(A))$
and similarly that
$fcolon Xto Y$ is surjective if and only if, for every $Csubset Y$, $C=f(f^{-1}(C))$
Hint for the $Rightarrow$ direction in the first statement: you have to prove that $f^{-1}(f(A))subset A$. Take $xin f^{-1}(f(A))$; then $f(x)in f(A)$. Apply injectivity.
answered Nov 26 at 22:24
egreg
178k1484200
Note that $f^{-1}(f(C))$ should be $f(f^{-1}(C))$.
The inclusion $Asubset f^{-1}(f(A))$ is true for every map and every subset of the map's domain.
Indeed, if $ain A$, then $f(a)in f(A)$ (by definition), which is to say that $ain f^{-1}(f(A))$.
Similarly, $f(f^{-1}(C))subset C$ holds for every map $f$ and every subset of the map's codomain. Indeed, if $yin f(f^{-1}(C))$, then $y=f(x)$, for some $xin f^{-1}(C)$. But then $f(x)in C$, which is to say $yin C$.
What you want to show is that
$fcolon Xto Y$ is injective if and only if, for every $Asubset X$, $A=f^{-1}(f(A))$
and similarly that
$fcolon Xto Y$ is surjective if and only if, for every $Csubset Y$, $C=f(f^{-1}(C))$
Hint for the $Rightarrow$ direction in the first statement: you have to prove that $f^{-1}(f(A))subset A$. Take $xin f^{-1}(f(A))$; then $f(x)in f(A)$. Apply injectivity.
answered Nov 26 at 22:24
egreg
178k1484200
answered Nov 26 at 22:24
egreg
178k1484200
answered Nov 26 at 22:24
egreg
178k1484200
answered Nov 26 at 22:24
egreg
178k1484200
178k1484200
I made the correction. The same hint holds for the surjectivity proposition?
– Armando Rosas
Nov 27 at 0:24
@ArmandoRosas Very similar. Don't forget to prove also the $Leftarrow$ directions.
– egreg
Nov 27 at 7:46
I have trouble with the $Leftarrow$ proof of surjectivity. I let c be in C , then c it's an element of $f(f^{-1}(C))$, can i directly say that f is surjective here? Because then $c=f(x)$ with $xin f^{-1}(C)subset X$, is it not?
– Armando Rosas
Nov 27 at 22:03
@ArmandoRosas For $C=Y$, $Y=f(f^{-1}(Y))=f(X)$.
– egreg
Nov 27 at 22:19
add a comment |
I made the correction. The same hint holds for the surjectivity proposition?
– Armando Rosas
Nov 27 at 0:24
@ArmandoRosas Very similar. Don't forget to prove also the $Leftarrow$ directions.
– egreg
Nov 27 at 7:46
I have trouble with the $Leftarrow$ proof of surjectivity. I let c be in C , then c it's an element of $f(f^{-1}(C))$, can i directly say that f is surjective here? Because then $c=f(x)$ with $xin f^{-1}(C)subset X$, is it not?
– Armando Rosas
Nov 27 at 22:03
@ArmandoRosas For $C=Y$, $Y=f(f^{-1}(Y))=f(X)$.
– egreg
Nov 27 at 22:19
I made the correction. The same hint holds for the surjectivity proposition?
– Armando Rosas
Nov 27 at 0:24
I made the correction. The same hint holds for the surjectivity proposition?
– Armando Rosas
Nov 27 at 0:24
@ArmandoRosas Very similar. Don't forget to prove also the $Leftarrow$ directions.
– egreg
Nov 27 at 7:46
@ArmandoRosas Very similar. Don't forget to prove also the $Leftarrow$ directions.
– egreg
Nov 27 at 7:46
I have trouble with the $Leftarrow$ proof of surjectivity. I let c be in C , then c it's an element of $f(f^{-1}(C))$, can i directly say that f is surjective here? Because then $c=f(x)$ with $xin f^{-1}(C)subset X$, is it not?
– Armando Rosas
Nov 27 at 22:03
I have trouble with the $Leftarrow$ proof of surjectivity. I let c be in C , then c it's an element of $f(f^{-1}(C))$, can i directly say that f is surjective here? Because then $c=f(x)$ with $xin f^{-1}(C)subset X$, is it not?
– Armando Rosas
Nov 27 at 22:03
@ArmandoRosas For $C=Y$, $Y=f(f^{-1}(Y))=f(X)$.
– egreg
Nov 27 at 22:19
@ArmandoRosas For $C=Y$, $Y=f(f^{-1}(Y))=f(X)$.
– egreg
Nov 27 at 22:19
add a comment |
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It is not enough that $A subset f^{-1}(f(A))$ for some $A subset X$, because let $f: {1, 2, 3} rightarrow {1, 2}$ with $f(1) = 1$, $f(2) = f(3) = 2$ and $A = {1}$ then $A subset f^{-1}(f(A))$ but f is not injective.
– BurningKarl
Nov 26 at 22:24
These properties are not true as stated. Are you sure this is what your homework says?
– Arnaud D.
Nov 26 at 22:25
@ArnaudD I'm not sure. That's how I have the exercises in the copy my teacher gave me.
– Armando Rosas
Nov 27 at 0:26