Idea behind this way of showing injectivity in a certain proof Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Every collection of disjoint non-empty open subsets of $mathbbR$ is countable?Special case of the Schroeder-Bernstein TheoremShowing injectivityWhy Is This Step Needed in Proving Bernstein-Schroeder?Doesn't “each x in X belongs to x/$mathscr E$” mean $bigcaplimits_x in X$ x/ $mathscr E$ = X? If not, is “each x in X” existential quantifier?Isn't x/E = y/ E ⇔ x E y deduced, not just x/E = y/ E ⇒ x E y from (a) x/ E≠ Ø, (b) x/E ∩ y/ E ≠ Ø ⇔xEy?Why $y/mathscr E$ is an element of $X/mathscr E$ when it's defined $X/mathscr E=,x/mathscr Emid xin X,$?Set $A$ partitionable into denumerable sets implies injection from $mathbbN$ to $A$Let $X$ be infinite. Then $X$ and $XtimesBbb N$ are equinumerousIf $X$ is infinite, then $X$ and $Xtimes X$ are equinumerous
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Idea behind this way of showing injectivity in a certain proof
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Every collection of disjoint non-empty open subsets of $mathbbR$ is countable?Special case of the Schroeder-Bernstein TheoremShowing injectivityWhy Is This Step Needed in Proving Bernstein-Schroeder?Doesn't “each x in X belongs to x/$mathscr E$” mean $bigcaplimits_x in X$ x/ $mathscr E$ = X? If not, is “each x in X” existential quantifier?Isn't x/E = y/ E ⇔ x E y deduced, not just x/E = y/ E ⇒ x E y from (a) x/ E≠ Ø, (b) x/E ∩ y/ E ≠ Ø ⇔xEy?Why $y/mathscr E$ is an element of $X/mathscr E$ when it's defined $X/mathscr E=,x/mathscr Emid xin X,$?Set $A$ partitionable into denumerable sets implies injection from $mathbbN$ to $A$Let $X$ be infinite. Then $X$ and $XtimesBbb N$ are equinumerousIf $X$ is infinite, then $X$ and $Xtimes X$ are equinumerous
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I would show injectivity of the function $h$ by showing $f(C) cap A-C = emptyset$. This is true, because $f(C) = f(bigcuplimits_ngeq 0 f^n(A-B)) = bigcuplimits_ngeq 1 f^n(A-B) subset C$ and $ A-C cap C = emptyset$. $f$ and $id_A-C$ are both injective and were are done.
But what is the author's idea ? In other words, how do
$A-B subset C$,$ f(C) subset C$,$f^m(A-B) cap f^n(A-B) = emptyset, n > m geq 0$,$f$ is one-one imply the injectivity of $h$ and why is this "better" here?
Source: McCleary - A First Course in Topology: Continuity and Dimension.
Actual proof taken from: Cox - A proof of the Schroeder-Bernstein Theorem, American Math. Monthly 75, 508
functions elementary-set-theory
asked Apr 1 at 17:57
QuantaurixQuantaurix
285
$endgroup$
|
show 2 more comments
$begingroup$
I would show injectivity of the function $h$ by showing $f(C) cap A-C = emptyset$. This is true, because $f(C) = f(bigcuplimits_ngeq 0 f^n(A-B)) = bigcuplimits_ngeq 1 f^n(A-B) subset C$ and $ A-C cap C = emptyset$. $f$ and $id_A-C$ are both injective and were are done.
But what is the author's idea ? In other words, how do
$A-B subset C$,$ f(C) subset C$,$f^m(A-B) cap f^n(A-B) = emptyset, n > m geq 0$,$f$ is one-one imply the injectivity of $h$ and why is this "better" here?
Source: McCleary - A First Course in Topology: Continuity and Dimension.
Actual proof taken from: Cox - A proof of the Schroeder-Bernstein Theorem, American Math. Monthly 75, 508
functions elementary-set-theory
asked Apr 1 at 17:57
QuantaurixQuantaurix
285
$endgroup$
$begingroup$
Can you precise the whole question? Is $f$ defined from $A$ to $A$ and $C subset A$?
$endgroup$
– freehumorist
Apr 1 at 18:14
$begingroup$
you probably overlooked the line "Lemma 1.9". $B subset A$, $f: A -> B$
$endgroup$
– Quantaurix
Apr 1 at 18:18
$begingroup$
Okay. You didn't even notice which Lemma from which book. Maybe I don't have the same book as yours?
$endgroup$
– freehumorist
Apr 1 at 18:19
1
$begingroup$
thanks, edited OP
$endgroup$
– Quantaurix
Apr 1 at 18:22
$begingroup$
As you are on Topology, the author wants you to use a topological approach. That's why the idea is better here. Thus, consider that (1) "the union of any number of open ball is an open ball too" (2) we strictly need to have $B subset A$ to be able to discuss injectivity for $f$. We must combine these two assumptions. Try to figure out yourself, and come back.
$endgroup$
– freehumorist
Apr 1 at 18:30
|
show 2 more comments
$begingroup$
I would show injectivity of the function $h$ by showing $f(C) cap A-C = emptyset$. This is true, because $f(C) = f(bigcuplimits_ngeq 0 f^n(A-B)) = bigcuplimits_ngeq 1 f^n(A-B) subset C$ and $ A-C cap C = emptyset$. $f$ and $id_A-C$ are both injective and were are done.
But what is the author's idea ? In other words, how do
$A-B subset C$,$ f(C) subset C$,$f^m(A-B) cap f^n(A-B) = emptyset, n > m geq 0$,$f$ is one-one imply the injectivity of $h$ and why is this "better" here?
Source: McCleary - A First Course in Topology: Continuity and Dimension.
Actual proof taken from: Cox - A proof of the Schroeder-Bernstein Theorem, American Math. Monthly 75, 508
functions elementary-set-theory
asked Apr 1 at 17:57
QuantaurixQuantaurix
285
$endgroup$
I would show injectivity of the function $h$ by showing $f(C) cap A-C = emptyset$. This is true, because $f(C) = f(bigcuplimits_ngeq 0 f^n(A-B)) = bigcuplimits_ngeq 1 f^n(A-B) subset C$ and $ A-C cap C = emptyset$. $f$ and $id_A-C$ are both injective and were are done.
But what is the author's idea ? In other words, how do
$A-B subset C$,$ f(C) subset C$,$f^m(A-B) cap f^n(A-B) = emptyset, n > m geq 0$,$f$ is one-one imply the injectivity of $h$ and why is this "better" here?
Source: McCleary - A First Course in Topology: Continuity and Dimension.
Actual proof taken from: Cox - A proof of the Schroeder-Bernstein Theorem, American Math. Monthly 75, 508
functions elementary-set-theory
functions elementary-set-theory
asked Apr 1 at 17:57
QuantaurixQuantaurix
285
asked Apr 1 at 17:57
QuantaurixQuantaurix
285
edited Apr 1 at 18:25
Quantaurix
asked Apr 1 at 17:57
QuantaurixQuantaurix
285
asked Apr 1 at 17:57
QuantaurixQuantaurix
285
asked Apr 1 at 17:57
QuantaurixQuantaurix
285
285
$begingroup$
Can you precise the whole question? Is $f$ defined from $A$ to $A$ and $C subset A$?
$endgroup$
– freehumorist
Apr 1 at 18:14
$begingroup$
you probably overlooked the line "Lemma 1.9". $B subset A$, $f: A -> B$
$endgroup$
– Quantaurix
Apr 1 at 18:18
$begingroup$
Okay. You didn't even notice which Lemma from which book. Maybe I don't have the same book as yours?
$endgroup$
– freehumorist
Apr 1 at 18:19
1
$begingroup$
thanks, edited OP
$endgroup$
– Quantaurix
Apr 1 at 18:22
$begingroup$
As you are on Topology, the author wants you to use a topological approach. That's why the idea is better here. Thus, consider that (1) "the union of any number of open ball is an open ball too" (2) we strictly need to have $B subset A$ to be able to discuss injectivity for $f$. We must combine these two assumptions. Try to figure out yourself, and come back.
$endgroup$
– freehumorist
Apr 1 at 18:30
|
show 2 more comments
$begingroup$
Can you precise the whole question? Is $f$ defined from $A$ to $A$ and $C subset A$?
$endgroup$
– freehumorist
Apr 1 at 18:14
$begingroup$
you probably overlooked the line "Lemma 1.9". $B subset A$, $f: A -> B$
$endgroup$
– Quantaurix
Apr 1 at 18:18
$begingroup$
Okay. You didn't even notice which Lemma from which book. Maybe I don't have the same book as yours?
$endgroup$
– freehumorist
Apr 1 at 18:19
1
$begingroup$
thanks, edited OP
$endgroup$
– Quantaurix
Apr 1 at 18:22
$begingroup$
As you are on Topology, the author wants you to use a topological approach. That's why the idea is better here. Thus, consider that (1) "the union of any number of open ball is an open ball too" (2) we strictly need to have $B subset A$ to be able to discuss injectivity for $f$. We must combine these two assumptions. Try to figure out yourself, and come back.
$endgroup$
– freehumorist
Apr 1 at 18:30
$begingroup$
Can you precise the whole question? Is $f$ defined from $A$ to $A$ and $C subset A$?
$endgroup$
– freehumorist
Apr 1 at 18:14
$begingroup$
Can you precise the whole question? Is $f$ defined from $A$ to $A$ and $C subset A$?
$endgroup$
– freehumorist
Apr 1 at 18:14
$begingroup$
you probably overlooked the line "Lemma 1.9". $B subset A$, $f: A -> B$
$endgroup$
– Quantaurix
Apr 1 at 18:18
$begingroup$
you probably overlooked the line "Lemma 1.9". $B subset A$, $f: A -> B$
$endgroup$
– Quantaurix
Apr 1 at 18:18
$begingroup$
Okay. You didn't even notice which Lemma from which book. Maybe I don't have the same book as yours?
$endgroup$
– freehumorist
Apr 1 at 18:19
$begingroup$
Okay. You didn't even notice which Lemma from which book. Maybe I don't have the same book as yours?
$endgroup$
– freehumorist
Apr 1 at 18:19
1
1
$begingroup$
thanks, edited OP
$endgroup$
– Quantaurix
Apr 1 at 18:22
$begingroup$
thanks, edited OP
$endgroup$
– Quantaurix
Apr 1 at 18:22
$begingroup$
As you are on Topology, the author wants you to use a topological approach. That's why the idea is better here. Thus, consider that (1) "the union of any number of open ball is an open ball too" (2) we strictly need to have $B subset A$ to be able to discuss injectivity for $f$. We must combine these two assumptions. Try to figure out yourself, and come back.
$endgroup$
– freehumorist
Apr 1 at 18:30
$begingroup$
As you are on Topology, the author wants you to use a topological approach. That's why the idea is better here. Thus, consider that (1) "the union of any number of open ball is an open ball too" (2) we strictly need to have $B subset A$ to be able to discuss injectivity for $f$. We must combine these two assumptions. Try to figure out yourself, and come back.
$endgroup$
– freehumorist
Apr 1 at 18:30
|
show 2 more comments
0
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$begingroup$
Can you precise the whole question? Is $f$ defined from $A$ to $A$ and $C subset A$?
$endgroup$
– freehumorist
Apr 1 at 18:14
$begingroup$
you probably overlooked the line "Lemma 1.9". $B subset A$, $f: A -> B$
$endgroup$
– Quantaurix
Apr 1 at 18:18
$begingroup$
Okay. You didn't even notice which Lemma from which book. Maybe I don't have the same book as yours?
$endgroup$
– freehumorist
Apr 1 at 18:19
1
$begingroup$
thanks, edited OP
$endgroup$
– Quantaurix
Apr 1 at 18:22
$begingroup$
As you are on Topology, the author wants you to use a topological approach. That's why the idea is better here. Thus, consider that (1) "the union of any number of open ball is an open ball too" (2) we strictly need to have $B subset A$ to be able to discuss injectivity for $f$. We must combine these two assumptions. Try to figure out yourself, and come back.
$endgroup$
– freehumorist
Apr 1 at 18:30