How to read $langle ~rangle$ in $langle F(a) | a in A rangle$?












0












$begingroup$


In Hrbacek and Jeck's book, Introduction to set theory, the symbol $langle~rangle$ is introduced for function representations.



I can read $F:A rightarrow B$, however, I can't read $langle F(a) | a in A rangle $,
$langle F_a | a in A rangle$, $langle F_a rangle _{ain A}$.



There are some examples:



$langle 2x-1 ~|~ x ~~ real rangle$,



$langle x^2 ~ |~ x ~~ real rangle$,



$langle frac{1}{x} ~|~ x~~real, ~~x neq 0 rangle$.



Each of $2x-1$, $x^2$, and $frac{1}{x}$ is not element of set. It is some value of a function.



In the second example, when $x=1$ or $x=-1$, the value of $x^2$ are both $1$.



Can I read the second example as "the collection of all $x$ squared such that x is real"?






elementary-set-theory notation







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    I don't think it's something you should read on its own. It's part of a larger notation, just like "$frac{phantom1}{phantom 2}$" isn't anything on its own but rather part of "$frac12$".
    $endgroup$
    – Arthur
    Dec 30 '18 at 7:50








  • 1




    $begingroup$
    It is simply a notation for "naming" functions (see page 24). Thus, it must be read : "the function $F$ with domain $A$".
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 30 '18 at 8:04






  • 1




    $begingroup$
    The collection of all $F_a$ indexed by a in A?
    $endgroup$
    – William Elliot
    Dec 30 '18 at 9:53






  • 1




    $begingroup$
    $langlerangle$
    $endgroup$
    – bof
    Dec 31 '18 at 11:47






  • 1




    $begingroup$
    If you right-click on an expression you can see the tex code.
    $endgroup$
    – bof
    Dec 31 '18 at 11:49
















0












$begingroup$


In Hrbacek and Jeck's book, Introduction to set theory, the symbol $langle~rangle$ is introduced for function representations.



I can read $F:A rightarrow B$, however, I can't read $langle F(a) | a in A rangle $,
$langle F_a | a in A rangle$, $langle F_a rangle _{ain A}$.



There are some examples:



$langle 2x-1 ~|~ x ~~ real rangle$,



$langle x^2 ~ |~ x ~~ real rangle$,



$langle frac{1}{x} ~|~ x~~real, ~~x neq 0 rangle$.



Each of $2x-1$, $x^2$, and $frac{1}{x}$ is not element of set. It is some value of a function.



In the second example, when $x=1$ or $x=-1$, the value of $x^2$ are both $1$.



Can I read the second example as "the collection of all $x$ squared such that x is real"?






elementary-set-theory notation







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    I don't think it's something you should read on its own. It's part of a larger notation, just like "$frac{phantom1}{phantom 2}$" isn't anything on its own but rather part of "$frac12$".
    $endgroup$
    – Arthur
    Dec 30 '18 at 7:50








  • 1




    $begingroup$
    It is simply a notation for "naming" functions (see page 24). Thus, it must be read : "the function $F$ with domain $A$".
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 30 '18 at 8:04






  • 1




    $begingroup$
    The collection of all $F_a$ indexed by a in A?
    $endgroup$
    – William Elliot
    Dec 30 '18 at 9:53






  • 1




    $begingroup$
    $langlerangle$
    $endgroup$
    – bof
    Dec 31 '18 at 11:47






  • 1




    $begingroup$
    If you right-click on an expression you can see the tex code.
    $endgroup$
    – bof
    Dec 31 '18 at 11:49














0












0








0





$begingroup$


In Hrbacek and Jeck's book, Introduction to set theory, the symbol $langle~rangle$ is introduced for function representations.



I can read $F:A rightarrow B$, however, I can't read $langle F(a) | a in A rangle $,
$langle F_a | a in A rangle$, $langle F_a rangle _{ain A}$.



There are some examples:



$langle 2x-1 ~|~ x ~~ real rangle$,



$langle x^2 ~ |~ x ~~ real rangle$,



$langle frac{1}{x} ~|~ x~~real, ~~x neq 0 rangle$.



Each of $2x-1$, $x^2$, and $frac{1}{x}$ is not element of set. It is some value of a function.



In the second example, when $x=1$ or $x=-1$, the value of $x^2$ are both $1$.



Can I read the second example as "the collection of all $x$ squared such that x is real"?






elementary-set-theory notation







share|cite|improve this question











$endgroup$




In Hrbacek and Jeck's book, Introduction to set theory, the symbol $langle~rangle$ is introduced for function representations.



I can read $F:A rightarrow B$, however, I can't read $langle F(a) | a in A rangle $,
$langle F_a | a in A rangle$, $langle F_a rangle _{ain A}$.



There are some examples:



$langle 2x-1 ~|~ x ~~ real rangle$,



$langle x^2 ~ |~ x ~~ real rangle$,



$langle frac{1}{x} ~|~ x~~real, ~~x neq 0 rangle$.



Each of $2x-1$, $x^2$, and $frac{1}{x}$ is not element of set. It is some value of a function.



In the second example, when $x=1$ or $x=-1$, the value of $x^2$ are both $1$.



Can I read the second example as "the collection of all $x$ squared such that x is real"?






elementary-set-theory notation




elementary-set-theory notation






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 31 '18 at 11:55







Doyun Nam

















asked Dec 30 '18 at 7:45









Doyun NamDoyun Nam

67119




67119








  • 1




    $begingroup$
    I don't think it's something you should read on its own. It's part of a larger notation, just like "$frac{phantom1}{phantom 2}$" isn't anything on its own but rather part of "$frac12$".
    $endgroup$
    – Arthur
    Dec 30 '18 at 7:50








  • 1




    $begingroup$
    It is simply a notation for "naming" functions (see page 24). Thus, it must be read : "the function $F$ with domain $A$".
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 30 '18 at 8:04






  • 1




    $begingroup$
    The collection of all $F_a$ indexed by a in A?
    $endgroup$
    – William Elliot
    Dec 30 '18 at 9:53






  • 1




    $begingroup$
    $langlerangle$
    $endgroup$
    – bof
    Dec 31 '18 at 11:47






  • 1




    $begingroup$
    If you right-click on an expression you can see the tex code.
    $endgroup$
    – bof
    Dec 31 '18 at 11:49














  • 1




    $begingroup$
    I don't think it's something you should read on its own. It's part of a larger notation, just like "$frac{phantom1}{phantom 2}$" isn't anything on its own but rather part of "$frac12$".
    $endgroup$
    – Arthur
    Dec 30 '18 at 7:50








  • 1




    $begingroup$
    It is simply a notation for "naming" functions (see page 24). Thus, it must be read : "the function $F$ with domain $A$".
    $endgroup$
    – Mauro ALLEGRANZA
    Dec 30 '18 at 8:04






  • 1




    $begingroup$
    The collection of all $F_a$ indexed by a in A?
    $endgroup$
    – William Elliot
    Dec 30 '18 at 9:53






  • 1




    $begingroup$
    $langlerangle$
    $endgroup$
    – bof
    Dec 31 '18 at 11:47






  • 1




    $begingroup$
    If you right-click on an expression you can see the tex code.
    $endgroup$
    – bof
    Dec 31 '18 at 11:49








1




1




$begingroup$
I don't think it's something you should read on its own. It's part of a larger notation, just like "$frac{phantom1}{phantom 2}$" isn't anything on its own but rather part of "$frac12$".
$endgroup$
– Arthur
Dec 30 '18 at 7:50






$begingroup$
I don't think it's something you should read on its own. It's part of a larger notation, just like "$frac{phantom1}{phantom 2}$" isn't anything on its own but rather part of "$frac12$".
$endgroup$
– Arthur
Dec 30 '18 at 7:50






1




1




$begingroup$
It is simply a notation for "naming" functions (see page 24). Thus, it must be read : "the function $F$ with domain $A$".
$endgroup$
– Mauro ALLEGRANZA
Dec 30 '18 at 8:04




$begingroup$
It is simply a notation for "naming" functions (see page 24). Thus, it must be read : "the function $F$ with domain $A$".
$endgroup$
– Mauro ALLEGRANZA
Dec 30 '18 at 8:04




1




1




$begingroup$
The collection of all $F_a$ indexed by a in A?
$endgroup$
– William Elliot
Dec 30 '18 at 9:53




$begingroup$
The collection of all $F_a$ indexed by a in A?
$endgroup$
– William Elliot
Dec 30 '18 at 9:53




1




1




$begingroup$
$langlerangle$
$endgroup$
– bof
Dec 31 '18 at 11:47




$begingroup$
$langlerangle$
$endgroup$
– bof
Dec 31 '18 at 11:47




1




1




$begingroup$
If you right-click on an expression you can see the tex code.
$endgroup$
– bof
Dec 31 '18 at 11:49




$begingroup$
If you right-click on an expression you can see the tex code.
$endgroup$
– bof
Dec 31 '18 at 11:49










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