Congruence Analytic Geometry Definition Question about Function where Its Input is a Set
$begingroup$
Below I am going to rewrite a definition which I found here.
$textbf{Definition:}$ Let $A, Bsubseteq mathbb{R}^n$. Then, $A$ and $B$ are said to be $textbf{congruent}$ (Euclidean metric) iff there exists an isometry (a distance preserving function) $f:mathbb{R}^nrightarrow mathbb{R}^n$ with $f(A)=B$.
$textbf{Question:}$ What does $f(A)=B$ mean here? $underline{Specifically}$, how do we define $f(A)$? I know $f(A)$ has a very specific meaning in Topology which I can't remember.
general-topology metric-spaces
edited Dec 26 '18 at 18:07
André 3000
12.8k22243
asked Dec 26 '18 at 14:51
W. G.W. G.
7061718
$endgroup$
add a comment |
$begingroup$
Below I am going to rewrite a definition which I found here.
$textbf{Definition:}$ Let $A, Bsubseteq mathbb{R}^n$. Then, $A$ and $B$ are said to be $textbf{congruent}$ (Euclidean metric) iff there exists an isometry (a distance preserving function) $f:mathbb{R}^nrightarrow mathbb{R}^n$ with $f(A)=B$.
$textbf{Question:}$ What does $f(A)=B$ mean here? $underline{Specifically}$, how do we define $f(A)$? I know $f(A)$ has a very specific meaning in Topology which I can't remember.
general-topology metric-spaces
edited Dec 26 '18 at 18:07
André 3000
12.8k22243
asked Dec 26 '18 at 14:51
W. G.W. G.
7061718
$endgroup$
1
$begingroup$
$f(A) = {f(a):ain A}$. This definition is not specific to topology, it's the same in all of mathematics.
$endgroup$
– Wojowu
Dec 26 '18 at 14:54
$begingroup$
@Wojowu You can write that in the answer too to get points (I know it will be a short answer, but that's exactly what I was looking for). Thank you!
$endgroup$
– W. G.
Dec 26 '18 at 14:56
add a comment |
$begingroup$
Below I am going to rewrite a definition which I found here.
$textbf{Definition:}$ Let $A, Bsubseteq mathbb{R}^n$. Then, $A$ and $B$ are said to be $textbf{congruent}$ (Euclidean metric) iff there exists an isometry (a distance preserving function) $f:mathbb{R}^nrightarrow mathbb{R}^n$ with $f(A)=B$.
$textbf{Question:}$ What does $f(A)=B$ mean here? $underline{Specifically}$, how do we define $f(A)$? I know $f(A)$ has a very specific meaning in Topology which I can't remember.
general-topology metric-spaces
edited Dec 26 '18 at 18:07
André 3000
12.8k22243
asked Dec 26 '18 at 14:51
W. G.W. G.
7061718
$endgroup$
Below I am going to rewrite a definition which I found here.
$textbf{Definition:}$ Let $A, Bsubseteq mathbb{R}^n$. Then, $A$ and $B$ are said to be $textbf{congruent}$ (Euclidean metric) iff there exists an isometry (a distance preserving function) $f:mathbb{R}^nrightarrow mathbb{R}^n$ with $f(A)=B$.
$textbf{Question:}$ What does $f(A)=B$ mean here? $underline{Specifically}$, how do we define $f(A)$? I know $f(A)$ has a very specific meaning in Topology which I can't remember.
general-topology metric-spaces
general-topology metric-spaces
edited Dec 26 '18 at 18:07
André 3000
12.8k22243
asked Dec 26 '18 at 14:51
W. G.W. G.
7061718
edited Dec 26 '18 at 18:07
André 3000
12.8k22243
asked Dec 26 '18 at 14:51
W. G.W. G.
7061718
edited Dec 26 '18 at 18:07
André 3000
12.8k22243
edited Dec 26 '18 at 18:07
André 3000
12.8k22243
edited Dec 26 '18 at 18:07
André 3000
12.8k22243
12.8k22243
asked Dec 26 '18 at 14:51
W. G.W. G.
7061718
asked Dec 26 '18 at 14:51
W. G.W. G.
7061718
asked Dec 26 '18 at 14:51
W. G.W. G.
7061718
7061718
1
$begingroup$
$f(A) = {f(a):ain A}$. This definition is not specific to topology, it's the same in all of mathematics.
$endgroup$
– Wojowu
Dec 26 '18 at 14:54
$begingroup$
@Wojowu You can write that in the answer too to get points (I know it will be a short answer, but that's exactly what I was looking for). Thank you!
$endgroup$
– W. G.
Dec 26 '18 at 14:56
add a comment |
1
$begingroup$
$f(A) = {f(a):ain A}$. This definition is not specific to topology, it's the same in all of mathematics.
$endgroup$
– Wojowu
Dec 26 '18 at 14:54
$begingroup$
@Wojowu You can write that in the answer too to get points (I know it will be a short answer, but that's exactly what I was looking for). Thank you!
$endgroup$
– W. G.
Dec 26 '18 at 14:56
1
1
$begingroup$
$f(A) = {f(a):ain A}$. This definition is not specific to topology, it's the same in all of mathematics.
$endgroup$
– Wojowu
Dec 26 '18 at 14:54
$begingroup$
$f(A) = {f(a):ain A}$. This definition is not specific to topology, it's the same in all of mathematics.
$endgroup$
– Wojowu
Dec 26 '18 at 14:54
$begingroup$
@Wojowu You can write that in the answer too to get points (I know it will be a short answer, but that's exactly what I was looking for). Thank you!
$endgroup$
– W. G.
Dec 26 '18 at 14:56
$begingroup$
@Wojowu You can write that in the answer too to get points (I know it will be a short answer, but that's exactly what I was looking for). Thank you!
$endgroup$
– W. G.
Dec 26 '18 at 14:56
add a comment |
1 Answer
1
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oldest
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$begingroup$
$f(A)$ is the image of the set $A$, which is the set of values which $f$ takes on elements of $A$. In symbols, $f(A)={f(a):ain A}$.
This definition is in no way specific to topology - it originates from set theory and is used all over the place in mathematics. Sometimes, to disambiguate between image of an element and an image of a subset, notation $f[A]$ is used instead.
answered Dec 26 '18 at 15:00
WojowuWojowu
19.3k23274
$endgroup$
add a comment |
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$begingroup$
$f(A)$ is the image of the set $A$, which is the set of values which $f$ takes on elements of $A$. In symbols, $f(A)={f(a):ain A}$.
This definition is in no way specific to topology - it originates from set theory and is used all over the place in mathematics. Sometimes, to disambiguate between image of an element and an image of a subset, notation $f[A]$ is used instead.
answered Dec 26 '18 at 15:00
WojowuWojowu
19.3k23274
$endgroup$
add a comment |
$begingroup$
$f(A)$ is the image of the set $A$, which is the set of values which $f$ takes on elements of $A$. In symbols, $f(A)={f(a):ain A}$.
This definition is in no way specific to topology - it originates from set theory and is used all over the place in mathematics. Sometimes, to disambiguate between image of an element and an image of a subset, notation $f[A]$ is used instead.
answered Dec 26 '18 at 15:00
WojowuWojowu
19.3k23274
$endgroup$
add a comment |
$begingroup$
$f(A)$ is the image of the set $A$, which is the set of values which $f$ takes on elements of $A$. In symbols, $f(A)={f(a):ain A}$.
This definition is in no way specific to topology - it originates from set theory and is used all over the place in mathematics. Sometimes, to disambiguate between image of an element and an image of a subset, notation $f[A]$ is used instead.
answered Dec 26 '18 at 15:00
WojowuWojowu
19.3k23274
$endgroup$
$f(A)$ is the image of the set $A$, which is the set of values which $f$ takes on elements of $A$. In symbols, $f(A)={f(a):ain A}$.
This definition is in no way specific to topology - it originates from set theory and is used all over the place in mathematics. Sometimes, to disambiguate between image of an element and an image of a subset, notation $f[A]$ is used instead.
answered Dec 26 '18 at 15:00
WojowuWojowu
19.3k23274
answered Dec 26 '18 at 15:00
WojowuWojowu
19.3k23274
answered Dec 26 '18 at 15:00
WojowuWojowu
19.3k23274
answered Dec 26 '18 at 15:00
WojowuWojowu
19.3k23274
19.3k23274
add a comment |
add a comment |
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$begingroup$
$f(A) = {f(a):ain A}$. This definition is not specific to topology, it's the same in all of mathematics.
$endgroup$
– Wojowu
Dec 26 '18 at 14:54
$begingroup$
@Wojowu You can write that in the answer too to get points (I know it will be a short answer, but that's exactly what I was looking for). Thank you!
$endgroup$
– W. G.
Dec 26 '18 at 14:56