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equivalent statements

dwsmith

Well-known member
Feb 1, 2012
1,673
Let $f:S\to T$ be a function.
Prove that the following statements are equivalent.


a
$f$ is one-to-one on $S$.




b
$f(A\cap B) = f(A)\cap f(B)$ for all subsets $A,B$ of $S$.




c
$f^{-1}[f(A)] = A$ for every subset $A$ of $S$.




d
For all disjoint subsets $A$ and $B$ of $S$, the images $f(A)$ and $f(B)$ are disjoint.

Having a tough time with this one.
 

Plato

Well-known member
MHB Math Helper
Jan 27, 2012
196
Let $f:S\to T$ be a function.
Prove that the following statements are equivalent.
a
$f$ is one-to-one on $S$.

b
$f(A\cap B) = f(A)\cap f(B)$ for all subsets $A,B$ of $S$.
For any function $f$ we have $f(A\cap B)\subseteq f(A)\cap f(B)$.
So suppose that $f$ is one-to-one and $t\in f(A)\cap f(B)$.
$\left( {\exists {a_t} \in A} \right)\left[ {f({a_t}) = t} \right]~\&~\left( {\exists {b_t} \in B} \right)\left[ {f({b_t}) = t} \right]$.
Use one-to-one to prove $t\in f(A\cap B)$.

Now start with $f(A\cap B) = f(A)\cap f(B)$ and continue.
 
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dwsmith

Well-known member
Feb 1, 2012
1,673
For any function $f$ we have $f(A\cap B)\subseteq f(A)\cap f(B)$.
So suppose that $f$ is one-to-one and $t\in f(A)\cap f(B)$.
$\left( {\exists {a_t} \in A} \right)\left[ {f({a_t}) = t} \right]~\&~\left( {\exists {b_t} \in B} \right)\left[ {f({b_t}) = t} \right]$.
Use one-to-one to prove $t\in f(A\cap B)$.

Now start with $f(A\cap B) = f(A)\cap f(B)$ and continue.

How can $f(A\cap B) = f(A)\cap f(B)$ be used to show c is true? I don't get it.
 

Plato

Well-known member
MHB Math Helper
Jan 27, 2012
196
How can $f(A\cap B) = f(A)\cap f(B)$ be used to show c is true? I don't get it.
I would show that $a) \Leftrightarrow c)$.
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
Note that $f^{-1}(f(A))\supseteq A$, so one only has to show $f^{-1}(f(A))\subseteq A$. Suppose $f^{-1}(f(A))=A\cup B$ where $A\cap B=\emptyset$. Using (b) show that $f(B)=\emptyset$ and therefore $B=\emptyset$.