Is it true that $Asubset f^{-1}(f(A))$ if and only if $f$ is injective?












1












$begingroup$


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.










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$endgroup$












  • $begingroup$
    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.
    $endgroup$
    – BurningKarl
    Nov 26 '18 at 22:24










  • $begingroup$
    These properties are not true as stated. Are you sure this is what your homework says?
    $endgroup$
    – Arnaud D.
    Nov 26 '18 at 22:25










  • $begingroup$
    @ArnaudD I'm not sure. That's how I have the exercises in the copy my teacher gave me.
    $endgroup$
    – Armando Rosas
    Nov 27 '18 at 0:26
















1












$begingroup$


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.










share|cite|improve this question











$endgroup$












  • $begingroup$
    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.
    $endgroup$
    – BurningKarl
    Nov 26 '18 at 22:24










  • $begingroup$
    These properties are not true as stated. Are you sure this is what your homework says?
    $endgroup$
    – Arnaud D.
    Nov 26 '18 at 22:25










  • $begingroup$
    @ArnaudD I'm not sure. That's how I have the exercises in the copy my teacher gave me.
    $endgroup$
    – Armando Rosas
    Nov 27 '18 at 0:26














1












1








1





$begingroup$


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.










share|cite|improve this question











$endgroup$




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






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edited Nov 27 '18 at 0:20







Armando Rosas

















asked Nov 26 '18 at 22:14









Armando RosasArmando Rosas

525




525












  • $begingroup$
    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.
    $endgroup$
    – BurningKarl
    Nov 26 '18 at 22:24










  • $begingroup$
    These properties are not true as stated. Are you sure this is what your homework says?
    $endgroup$
    – Arnaud D.
    Nov 26 '18 at 22:25










  • $begingroup$
    @ArnaudD I'm not sure. That's how I have the exercises in the copy my teacher gave me.
    $endgroup$
    – Armando Rosas
    Nov 27 '18 at 0:26


















  • $begingroup$
    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.
    $endgroup$
    – BurningKarl
    Nov 26 '18 at 22:24










  • $begingroup$
    These properties are not true as stated. Are you sure this is what your homework says?
    $endgroup$
    – Arnaud D.
    Nov 26 '18 at 22:25










  • $begingroup$
    @ArnaudD I'm not sure. That's how I have the exercises in the copy my teacher gave me.
    $endgroup$
    – Armando Rosas
    Nov 27 '18 at 0:26
















$begingroup$
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.
$endgroup$
– BurningKarl
Nov 26 '18 at 22:24




$begingroup$
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.
$endgroup$
– BurningKarl
Nov 26 '18 at 22:24












$begingroup$
These properties are not true as stated. Are you sure this is what your homework says?
$endgroup$
– Arnaud D.
Nov 26 '18 at 22:25




$begingroup$
These properties are not true as stated. Are you sure this is what your homework says?
$endgroup$
– Arnaud D.
Nov 26 '18 at 22:25












$begingroup$
@ArnaudD I'm not sure. That's how I have the exercises in the copy my teacher gave me.
$endgroup$
– Armando Rosas
Nov 27 '18 at 0:26




$begingroup$
@ArnaudD I'm not sure. That's how I have the exercises in the copy my teacher gave me.
$endgroup$
– Armando Rosas
Nov 27 '18 at 0:26










1 Answer
1






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oldest

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4












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I made the correction. The same hint holds for the surjectivity proposition?
    $endgroup$
    – Armando Rosas
    Nov 27 '18 at 0:24










  • $begingroup$
    @ArmandoRosas Very similar. Don't forget to prove also the $Leftarrow$ directions.
    $endgroup$
    – egreg
    Nov 27 '18 at 7:46










  • $begingroup$
    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?
    $endgroup$
    – Armando Rosas
    Nov 27 '18 at 22:03










  • $begingroup$
    @ArmandoRosas For $C=Y$, $Y=f(f^{-1}(Y))=f(X)$.
    $endgroup$
    – egreg
    Nov 27 '18 at 22:19











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1 Answer
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1 Answer
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active

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active

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4












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I made the correction. The same hint holds for the surjectivity proposition?
    $endgroup$
    – Armando Rosas
    Nov 27 '18 at 0:24










  • $begingroup$
    @ArmandoRosas Very similar. Don't forget to prove also the $Leftarrow$ directions.
    $endgroup$
    – egreg
    Nov 27 '18 at 7:46










  • $begingroup$
    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?
    $endgroup$
    – Armando Rosas
    Nov 27 '18 at 22:03










  • $begingroup$
    @ArmandoRosas For $C=Y$, $Y=f(f^{-1}(Y))=f(X)$.
    $endgroup$
    – egreg
    Nov 27 '18 at 22:19
















4












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    I made the correction. The same hint holds for the surjectivity proposition?
    $endgroup$
    – Armando Rosas
    Nov 27 '18 at 0:24










  • $begingroup$
    @ArmandoRosas Very similar. Don't forget to prove also the $Leftarrow$ directions.
    $endgroup$
    – egreg
    Nov 27 '18 at 7:46










  • $begingroup$
    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?
    $endgroup$
    – Armando Rosas
    Nov 27 '18 at 22:03










  • $begingroup$
    @ArmandoRosas For $C=Y$, $Y=f(f^{-1}(Y))=f(X)$.
    $endgroup$
    – egreg
    Nov 27 '18 at 22:19














4












4








4





$begingroup$

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.






share|cite|improve this answer









$endgroup$



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.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 26 '18 at 22:24









egregegreg

180k1485202




180k1485202












  • $begingroup$
    I made the correction. The same hint holds for the surjectivity proposition?
    $endgroup$
    – Armando Rosas
    Nov 27 '18 at 0:24










  • $begingroup$
    @ArmandoRosas Very similar. Don't forget to prove also the $Leftarrow$ directions.
    $endgroup$
    – egreg
    Nov 27 '18 at 7:46










  • $begingroup$
    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?
    $endgroup$
    – Armando Rosas
    Nov 27 '18 at 22:03










  • $begingroup$
    @ArmandoRosas For $C=Y$, $Y=f(f^{-1}(Y))=f(X)$.
    $endgroup$
    – egreg
    Nov 27 '18 at 22:19


















  • $begingroup$
    I made the correction. The same hint holds for the surjectivity proposition?
    $endgroup$
    – Armando Rosas
    Nov 27 '18 at 0:24










  • $begingroup$
    @ArmandoRosas Very similar. Don't forget to prove also the $Leftarrow$ directions.
    $endgroup$
    – egreg
    Nov 27 '18 at 7:46










  • $begingroup$
    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?
    $endgroup$
    – Armando Rosas
    Nov 27 '18 at 22:03










  • $begingroup$
    @ArmandoRosas For $C=Y$, $Y=f(f^{-1}(Y))=f(X)$.
    $endgroup$
    – egreg
    Nov 27 '18 at 22:19
















$begingroup$
I made the correction. The same hint holds for the surjectivity proposition?
$endgroup$
– Armando Rosas
Nov 27 '18 at 0:24




$begingroup$
I made the correction. The same hint holds for the surjectivity proposition?
$endgroup$
– Armando Rosas
Nov 27 '18 at 0:24












$begingroup$
@ArmandoRosas Very similar. Don't forget to prove also the $Leftarrow$ directions.
$endgroup$
– egreg
Nov 27 '18 at 7:46




$begingroup$
@ArmandoRosas Very similar. Don't forget to prove also the $Leftarrow$ directions.
$endgroup$
– egreg
Nov 27 '18 at 7:46












$begingroup$
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?
$endgroup$
– Armando Rosas
Nov 27 '18 at 22:03




$begingroup$
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?
$endgroup$
– Armando Rosas
Nov 27 '18 at 22:03












$begingroup$
@ArmandoRosas For $C=Y$, $Y=f(f^{-1}(Y))=f(X)$.
$endgroup$
– egreg
Nov 27 '18 at 22:19




$begingroup$
@ArmandoRosas For $C=Y$, $Y=f(f^{-1}(Y))=f(X)$.
$endgroup$
– egreg
Nov 27 '18 at 22:19


















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