Notation for extracting value out of single element set












2














In some part of a document I am writing, I first define a set



$$
S : A to mathcal P(B) \
S(a) = { text{complicated expression here using $a$} }
$$



Later, I prove that $forall a,~exists x, ~S(a) = { x }$.



So, I want to now construct some notation for referring to the set with a single element ${ x } subset B$. Something like



$$unset(S(a))$$



where $unset$ is some valid definition / notation.






elementary-set-theory notation







share|cite|improve this question



























    2














    In some part of a document I am writing, I first define a set



    $$
    S : A to mathcal P(B) \
    S(a) = { text{complicated expression here using $a$} }
    $$



    Later, I prove that $forall a,~exists x, ~S(a) = { x }$.



    So, I want to now construct some notation for referring to the set with a single element ${ x } subset B$. Something like



    $$unset(S(a))$$



    where $unset$ is some valid definition / notation.






    elementary-set-theory notation







    share|cite|improve this question

























      2












      2








      2







      In some part of a document I am writing, I first define a set



      $$
      S : A to mathcal P(B) \
      S(a) = { text{complicated expression here using $a$} }
      $$



      Later, I prove that $forall a,~exists x, ~S(a) = { x }$.



      So, I want to now construct some notation for referring to the set with a single element ${ x } subset B$. Something like



      $$unset(S(a))$$



      where $unset$ is some valid definition / notation.






      elementary-set-theory notation







      share|cite|improve this question













      In some part of a document I am writing, I first define a set



      $$
      S : A to mathcal P(B) \
      S(a) = { text{complicated expression here using $a$} }
      $$



      Later, I prove that $forall a,~exists x, ~S(a) = { x }$.



      So, I want to now construct some notation for referring to the set with a single element ${ x } subset B$. Something like



      $$unset(S(a))$$



      where $unset$ is some valid definition / notation.






      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










      asked 1 hour ago









      Siddharth Bhat

      2,8481918




      2,8481918






















          3 Answers
          3






          active

          oldest

          votes


















          2














          There is no commonly used notation specifically for this, so you shouldn't hesitate to just make up your own. One way you can express it with standard notation is with the symbol $bigcup$: if $S$ is a set, then $bigcup S$ denotes the union of all the elements of $S$, so $bigcup {x}=x$. This is kind of a "hack" though and is not something you can expect your readers to effortlessly understand. (Mathematically literate readers will figure it out, but it may take them a little work.)



          Ultimately, the purpose of notation is to communicate, so you should pick your notation to communicate clearly and not be afraid to use words instead of notation if that would be clearer. I would probably recommend instead just writing something like:




          We write $s(a)$ for the unique element of the set $S(a)$.




          There's nothing special about the choice of $s$ as the function name for this; it's a reasonable choice to use to remind the reader that it is related to $S$ but there's nothing wrong with using a different name if you have a better reason.






          share|cite|improve this answer























          • Thank you, I suppose I'll just do that then :) I was hoping for some existing notation, but oh well.
            – Siddharth Bhat
            1 hour ago










          • I have doubts regarding the validity of blindly using this notation. I think that implicitly the union must be over a collection of sets. If not then what the heck would $2cup {3}$ possibly mean?
            – MPW
            23 mins ago










          • @MPW Remember that $2={{},{{}}}$, so it's perfectly meaningful. :P (At least, if we're working in set theory.) More to the point, as long as the OP doesn't write "$bigcup x$" when $x$ isn't a set, they'll be fine. And actually, even outside of ZFC you can arguably make sense of all such expressions: e.g. "$2cup{3}$" is by definition ${x: xin 2$ or $xin{3}}$, and if $2$ has no elements this is just ${3}$. :P
            – Noah Schweber
            8 mins ago



















          1














          It is just the union:



          $$bigcup S(a) = bigcup{x} = x$$



          Indeed, the union of a set $A$ is defined with $$c in bigcup A iff exists D in A text{ such that } c in D$$



          For two sets we have the more familiar notation $A cup B = bigcup {A,B}$.






          share|cite|improve this answer

















          • 1




            This is true but I wouldn't necessarily recommend actually using this notation--it is not particularly evocative of the desired meaning here and may confuse some readers.
            – Eric Wofsey
            1 hour ago










          • @EricWofsey Agreed, I wouldn't use any kind of "$text{unset}$" function. It would be best to state that $S(a) = {x}$ and then just use $x$.
            – mechanodroid
            1 hour ago












          • This is neat, I would not have thought of that! However, I'm accepting Eric's answer, since I believe it answers the question's spirit better, by suggesting some kind of wording.
            – Siddharth Bhat
            1 hour ago



















          0














          Since it looks like you’re going to have to invent your own notation, why not just simply $hat{a}$ ? Why bring $S$ into the notation at all?






          share|cite|improve this answer





















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            3 Answers
            3






            active

            oldest

            votes








            3 Answers
            3






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            2














            There is no commonly used notation specifically for this, so you shouldn't hesitate to just make up your own. One way you can express it with standard notation is with the symbol $bigcup$: if $S$ is a set, then $bigcup S$ denotes the union of all the elements of $S$, so $bigcup {x}=x$. This is kind of a "hack" though and is not something you can expect your readers to effortlessly understand. (Mathematically literate readers will figure it out, but it may take them a little work.)



            Ultimately, the purpose of notation is to communicate, so you should pick your notation to communicate clearly and not be afraid to use words instead of notation if that would be clearer. I would probably recommend instead just writing something like:




            We write $s(a)$ for the unique element of the set $S(a)$.




            There's nothing special about the choice of $s$ as the function name for this; it's a reasonable choice to use to remind the reader that it is related to $S$ but there's nothing wrong with using a different name if you have a better reason.






            share|cite|improve this answer























            • Thank you, I suppose I'll just do that then :) I was hoping for some existing notation, but oh well.
              – Siddharth Bhat
              1 hour ago










            • I have doubts regarding the validity of blindly using this notation. I think that implicitly the union must be over a collection of sets. If not then what the heck would $2cup {3}$ possibly mean?
              – MPW
              23 mins ago










            • @MPW Remember that $2={{},{{}}}$, so it's perfectly meaningful. :P (At least, if we're working in set theory.) More to the point, as long as the OP doesn't write "$bigcup x$" when $x$ isn't a set, they'll be fine. And actually, even outside of ZFC you can arguably make sense of all such expressions: e.g. "$2cup{3}$" is by definition ${x: xin 2$ or $xin{3}}$, and if $2$ has no elements this is just ${3}$. :P
              – Noah Schweber
              8 mins ago
















            2














            There is no commonly used notation specifically for this, so you shouldn't hesitate to just make up your own. One way you can express it with standard notation is with the symbol $bigcup$: if $S$ is a set, then $bigcup S$ denotes the union of all the elements of $S$, so $bigcup {x}=x$. This is kind of a "hack" though and is not something you can expect your readers to effortlessly understand. (Mathematically literate readers will figure it out, but it may take them a little work.)



            Ultimately, the purpose of notation is to communicate, so you should pick your notation to communicate clearly and not be afraid to use words instead of notation if that would be clearer. I would probably recommend instead just writing something like:




            We write $s(a)$ for the unique element of the set $S(a)$.




            There's nothing special about the choice of $s$ as the function name for this; it's a reasonable choice to use to remind the reader that it is related to $S$ but there's nothing wrong with using a different name if you have a better reason.






            share|cite|improve this answer























            • Thank you, I suppose I'll just do that then :) I was hoping for some existing notation, but oh well.
              – Siddharth Bhat
              1 hour ago










            • I have doubts regarding the validity of blindly using this notation. I think that implicitly the union must be over a collection of sets. If not then what the heck would $2cup {3}$ possibly mean?
              – MPW
              23 mins ago










            • @MPW Remember that $2={{},{{}}}$, so it's perfectly meaningful. :P (At least, if we're working in set theory.) More to the point, as long as the OP doesn't write "$bigcup x$" when $x$ isn't a set, they'll be fine. And actually, even outside of ZFC you can arguably make sense of all such expressions: e.g. "$2cup{3}$" is by definition ${x: xin 2$ or $xin{3}}$, and if $2$ has no elements this is just ${3}$. :P
              – Noah Schweber
              8 mins ago














            2












            2








            2






            There is no commonly used notation specifically for this, so you shouldn't hesitate to just make up your own. One way you can express it with standard notation is with the symbol $bigcup$: if $S$ is a set, then $bigcup S$ denotes the union of all the elements of $S$, so $bigcup {x}=x$. This is kind of a "hack" though and is not something you can expect your readers to effortlessly understand. (Mathematically literate readers will figure it out, but it may take them a little work.)



            Ultimately, the purpose of notation is to communicate, so you should pick your notation to communicate clearly and not be afraid to use words instead of notation if that would be clearer. I would probably recommend instead just writing something like:




            We write $s(a)$ for the unique element of the set $S(a)$.




            There's nothing special about the choice of $s$ as the function name for this; it's a reasonable choice to use to remind the reader that it is related to $S$ but there's nothing wrong with using a different name if you have a better reason.






            share|cite|improve this answer














            There is no commonly used notation specifically for this, so you shouldn't hesitate to just make up your own. One way you can express it with standard notation is with the symbol $bigcup$: if $S$ is a set, then $bigcup S$ denotes the union of all the elements of $S$, so $bigcup {x}=x$. This is kind of a "hack" though and is not something you can expect your readers to effortlessly understand. (Mathematically literate readers will figure it out, but it may take them a little work.)



            Ultimately, the purpose of notation is to communicate, so you should pick your notation to communicate clearly and not be afraid to use words instead of notation if that would be clearer. I would probably recommend instead just writing something like:




            We write $s(a)$ for the unique element of the set $S(a)$.




            There's nothing special about the choice of $s$ as the function name for this; it's a reasonable choice to use to remind the reader that it is related to $S$ but there's nothing wrong with using a different name if you have a better reason.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited 59 mins ago

























            answered 1 hour ago









            Eric Wofsey

            179k12204331




            179k12204331












            • Thank you, I suppose I'll just do that then :) I was hoping for some existing notation, but oh well.
              – Siddharth Bhat
              1 hour ago










            • I have doubts regarding the validity of blindly using this notation. I think that implicitly the union must be over a collection of sets. If not then what the heck would $2cup {3}$ possibly mean?
              – MPW
              23 mins ago










            • @MPW Remember that $2={{},{{}}}$, so it's perfectly meaningful. :P (At least, if we're working in set theory.) More to the point, as long as the OP doesn't write "$bigcup x$" when $x$ isn't a set, they'll be fine. And actually, even outside of ZFC you can arguably make sense of all such expressions: e.g. "$2cup{3}$" is by definition ${x: xin 2$ or $xin{3}}$, and if $2$ has no elements this is just ${3}$. :P
              – Noah Schweber
              8 mins ago


















            • Thank you, I suppose I'll just do that then :) I was hoping for some existing notation, but oh well.
              – Siddharth Bhat
              1 hour ago










            • I have doubts regarding the validity of blindly using this notation. I think that implicitly the union must be over a collection of sets. If not then what the heck would $2cup {3}$ possibly mean?
              – MPW
              23 mins ago










            • @MPW Remember that $2={{},{{}}}$, so it's perfectly meaningful. :P (At least, if we're working in set theory.) More to the point, as long as the OP doesn't write "$bigcup x$" when $x$ isn't a set, they'll be fine. And actually, even outside of ZFC you can arguably make sense of all such expressions: e.g. "$2cup{3}$" is by definition ${x: xin 2$ or $xin{3}}$, and if $2$ has no elements this is just ${3}$. :P
              – Noah Schweber
              8 mins ago
















            Thank you, I suppose I'll just do that then :) I was hoping for some existing notation, but oh well.
            – Siddharth Bhat
            1 hour ago




            Thank you, I suppose I'll just do that then :) I was hoping for some existing notation, but oh well.
            – Siddharth Bhat
            1 hour ago












            I have doubts regarding the validity of blindly using this notation. I think that implicitly the union must be over a collection of sets. If not then what the heck would $2cup {3}$ possibly mean?
            – MPW
            23 mins ago




            I have doubts regarding the validity of blindly using this notation. I think that implicitly the union must be over a collection of sets. If not then what the heck would $2cup {3}$ possibly mean?
            – MPW
            23 mins ago












            @MPW Remember that $2={{},{{}}}$, so it's perfectly meaningful. :P (At least, if we're working in set theory.) More to the point, as long as the OP doesn't write "$bigcup x$" when $x$ isn't a set, they'll be fine. And actually, even outside of ZFC you can arguably make sense of all such expressions: e.g. "$2cup{3}$" is by definition ${x: xin 2$ or $xin{3}}$, and if $2$ has no elements this is just ${3}$. :P
            – Noah Schweber
            8 mins ago




            @MPW Remember that $2={{},{{}}}$, so it's perfectly meaningful. :P (At least, if we're working in set theory.) More to the point, as long as the OP doesn't write "$bigcup x$" when $x$ isn't a set, they'll be fine. And actually, even outside of ZFC you can arguably make sense of all such expressions: e.g. "$2cup{3}$" is by definition ${x: xin 2$ or $xin{3}}$, and if $2$ has no elements this is just ${3}$. :P
            – Noah Schweber
            8 mins ago











            1














            It is just the union:



            $$bigcup S(a) = bigcup{x} = x$$



            Indeed, the union of a set $A$ is defined with $$c in bigcup A iff exists D in A text{ such that } c in D$$



            For two sets we have the more familiar notation $A cup B = bigcup {A,B}$.






            share|cite|improve this answer

















            • 1




              This is true but I wouldn't necessarily recommend actually using this notation--it is not particularly evocative of the desired meaning here and may confuse some readers.
              – Eric Wofsey
              1 hour ago










            • @EricWofsey Agreed, I wouldn't use any kind of "$text{unset}$" function. It would be best to state that $S(a) = {x}$ and then just use $x$.
              – mechanodroid
              1 hour ago












            • This is neat, I would not have thought of that! However, I'm accepting Eric's answer, since I believe it answers the question's spirit better, by suggesting some kind of wording.
              – Siddharth Bhat
              1 hour ago
















            1














            It is just the union:



            $$bigcup S(a) = bigcup{x} = x$$



            Indeed, the union of a set $A$ is defined with $$c in bigcup A iff exists D in A text{ such that } c in D$$



            For two sets we have the more familiar notation $A cup B = bigcup {A,B}$.






            share|cite|improve this answer

















            • 1




              This is true but I wouldn't necessarily recommend actually using this notation--it is not particularly evocative of the desired meaning here and may confuse some readers.
              – Eric Wofsey
              1 hour ago










            • @EricWofsey Agreed, I wouldn't use any kind of "$text{unset}$" function. It would be best to state that $S(a) = {x}$ and then just use $x$.
              – mechanodroid
              1 hour ago












            • This is neat, I would not have thought of that! However, I'm accepting Eric's answer, since I believe it answers the question's spirit better, by suggesting some kind of wording.
              – Siddharth Bhat
              1 hour ago














            1












            1








            1






            It is just the union:



            $$bigcup S(a) = bigcup{x} = x$$



            Indeed, the union of a set $A$ is defined with $$c in bigcup A iff exists D in A text{ such that } c in D$$



            For two sets we have the more familiar notation $A cup B = bigcup {A,B}$.






            share|cite|improve this answer












            It is just the union:



            $$bigcup S(a) = bigcup{x} = x$$



            Indeed, the union of a set $A$ is defined with $$c in bigcup A iff exists D in A text{ such that } c in D$$



            For two sets we have the more familiar notation $A cup B = bigcup {A,B}$.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 1 hour ago









            mechanodroid

            26.1k62245




            26.1k62245








            • 1




              This is true but I wouldn't necessarily recommend actually using this notation--it is not particularly evocative of the desired meaning here and may confuse some readers.
              – Eric Wofsey
              1 hour ago










            • @EricWofsey Agreed, I wouldn't use any kind of "$text{unset}$" function. It would be best to state that $S(a) = {x}$ and then just use $x$.
              – mechanodroid
              1 hour ago












            • This is neat, I would not have thought of that! However, I'm accepting Eric's answer, since I believe it answers the question's spirit better, by suggesting some kind of wording.
              – Siddharth Bhat
              1 hour ago














            • 1




              This is true but I wouldn't necessarily recommend actually using this notation--it is not particularly evocative of the desired meaning here and may confuse some readers.
              – Eric Wofsey
              1 hour ago










            • @EricWofsey Agreed, I wouldn't use any kind of "$text{unset}$" function. It would be best to state that $S(a) = {x}$ and then just use $x$.
              – mechanodroid
              1 hour ago












            • This is neat, I would not have thought of that! However, I'm accepting Eric's answer, since I believe it answers the question's spirit better, by suggesting some kind of wording.
              – Siddharth Bhat
              1 hour ago








            1




            1




            This is true but I wouldn't necessarily recommend actually using this notation--it is not particularly evocative of the desired meaning here and may confuse some readers.
            – Eric Wofsey
            1 hour ago




            This is true but I wouldn't necessarily recommend actually using this notation--it is not particularly evocative of the desired meaning here and may confuse some readers.
            – Eric Wofsey
            1 hour ago












            @EricWofsey Agreed, I wouldn't use any kind of "$text{unset}$" function. It would be best to state that $S(a) = {x}$ and then just use $x$.
            – mechanodroid
            1 hour ago






            @EricWofsey Agreed, I wouldn't use any kind of "$text{unset}$" function. It would be best to state that $S(a) = {x}$ and then just use $x$.
            – mechanodroid
            1 hour ago














            This is neat, I would not have thought of that! However, I'm accepting Eric's answer, since I believe it answers the question's spirit better, by suggesting some kind of wording.
            – Siddharth Bhat
            1 hour ago




            This is neat, I would not have thought of that! However, I'm accepting Eric's answer, since I believe it answers the question's spirit better, by suggesting some kind of wording.
            – Siddharth Bhat
            1 hour ago











            0














            Since it looks like you’re going to have to invent your own notation, why not just simply $hat{a}$ ? Why bring $S$ into the notation at all?






            share|cite|improve this answer


























              0














              Since it looks like you’re going to have to invent your own notation, why not just simply $hat{a}$ ? Why bring $S$ into the notation at all?






              share|cite|improve this answer
























                0












                0








                0






                Since it looks like you’re going to have to invent your own notation, why not just simply $hat{a}$ ? Why bring $S$ into the notation at all?






                share|cite|improve this answer












                Since it looks like you’re going to have to invent your own notation, why not just simply $hat{a}$ ? Why bring $S$ into the notation at all?







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 31 mins ago









                MPW

                29.8k12056




                29.8k12056






























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                    Sidney Franklin

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                    Questa voce sull'argomento registi statunitensi è solo un abbozzo . Contribuisci a migliorarla secondo le convenzioni di Wikipedia. Sidney Franklin, foto di Carl Van Vechten Oscar alla memoria Irving G. Thalberg 1943 Sidney Franklin , all'anagrafe Sidney Arnold Franklin , (San Francisco, 21 marzo 1893 – Santa Monica, 18 maggio 1972), è stato un regista, produttore cinematografico e attore statunitense. Il 20 giugno 1963, si sposò con l'attrice australiana Enid Bennett. Il matrimonio durò fino alla morte dell'attrice, il 14 maggio 1969 [1] Indice 1 Filmografia 1.1 Regista 1.2 Aiuto regia (parziale) 1.3 Produttore 1.4 Attore (parziale) 2 Note 3 Altri progetti 4 Collegamenti esterni Filmografia | Regista | The Baby , co-regia Chester M. Franklin (1915) The Rivals , co-regia Chester M. Franklin (1915) Little Dick's First Case , co-regia Chester M. Franklin (1915) Her Filmland Hero , co-...
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