Let $a in A and U in C such that a in U $. Prove that $[a] = U $.












0














If you refer to this link and the question, it's this same question, this is the last part of the 4 part question .



The equivalence relation $sim$ is defined as:
$$textsf{For }x,yin A,xsim ytextsf{ if and only if there exists }Uin Ctextsf{ such that }xin Utextsf{ and }yin U$$



I have to prove that $[a] = U $ given that: $$a in Atextsf{ and }U in Ctextsf{ such that }a in U$$ I mentioned that the definition of a partition says that: $$textsf{For all }a in Atextsf{ there exists }U in Ctextsf{ such that }a in U$$... but I'm not sure where to go from here.






equivalence-relations set-partition







share|cite|improve this question




















  • 1




    First, what is $[a]$?
    – Graham Kemp
    Nov 30 '18 at 1:49










  • $[a]$ is an equivalence class @GrahamKemp
    – Claire
    Nov 30 '18 at 1:50










  • For what equivalence relation?
    – Graham Kemp
    Nov 30 '18 at 1:51












  • I assume you are trying to show a step in the problem of showing that partitions are effectively equivalence relations and vice versa on a set. I further assume that you have some specific equivalence relation $sim$ for your specific partition $C$. You are likely intended to show that the equivalence class $[a]$ is equal to the part $U$. There is a big difference between "$[a]in U$" and "$[a]=U$."
    – JMoravitz
    Nov 30 '18 at 1:51








  • 1




    Yes I do, if you refer to this link and the question, it's this same question, this is the last part of the 4 part question. math.stackexchange.com/q/3019461/603668
    – Claire
    Nov 30 '18 at 1:53
















0














If you refer to this link and the question, it's this same question, this is the last part of the 4 part question .



The equivalence relation $sim$ is defined as:
$$textsf{For }x,yin A,xsim ytextsf{ if and only if there exists }Uin Ctextsf{ such that }xin Utextsf{ and }yin U$$



I have to prove that $[a] = U $ given that: $$a in Atextsf{ and }U in Ctextsf{ such that }a in U$$ I mentioned that the definition of a partition says that: $$textsf{For all }a in Atextsf{ there exists }U in Ctextsf{ such that }a in U$$... but I'm not sure where to go from here.






equivalence-relations set-partition







share|cite|improve this question




















  • 1




    First, what is $[a]$?
    – Graham Kemp
    Nov 30 '18 at 1:49










  • $[a]$ is an equivalence class @GrahamKemp
    – Claire
    Nov 30 '18 at 1:50










  • For what equivalence relation?
    – Graham Kemp
    Nov 30 '18 at 1:51












  • I assume you are trying to show a step in the problem of showing that partitions are effectively equivalence relations and vice versa on a set. I further assume that you have some specific equivalence relation $sim$ for your specific partition $C$. You are likely intended to show that the equivalence class $[a]$ is equal to the part $U$. There is a big difference between "$[a]in U$" and "$[a]=U$."
    – JMoravitz
    Nov 30 '18 at 1:51








  • 1




    Yes I do, if you refer to this link and the question, it's this same question, this is the last part of the 4 part question. math.stackexchange.com/q/3019461/603668
    – Claire
    Nov 30 '18 at 1:53














0












0








0







If you refer to this link and the question, it's this same question, this is the last part of the 4 part question .



The equivalence relation $sim$ is defined as:
$$textsf{For }x,yin A,xsim ytextsf{ if and only if there exists }Uin Ctextsf{ such that }xin Utextsf{ and }yin U$$



I have to prove that $[a] = U $ given that: $$a in Atextsf{ and }U in Ctextsf{ such that }a in U$$ I mentioned that the definition of a partition says that: $$textsf{For all }a in Atextsf{ there exists }U in Ctextsf{ such that }a in U$$... but I'm not sure where to go from here.






equivalence-relations set-partition







share|cite|improve this question















If you refer to this link and the question, it's this same question, this is the last part of the 4 part question .



The equivalence relation $sim$ is defined as:
$$textsf{For }x,yin A,xsim ytextsf{ if and only if there exists }Uin Ctextsf{ such that }xin Utextsf{ and }yin U$$



I have to prove that $[a] = U $ given that: $$a in Atextsf{ and }U in Ctextsf{ such that }a in U$$ I mentioned that the definition of a partition says that: $$textsf{For all }a in Atextsf{ there exists }U in Ctextsf{ such that }a in U$$... but I'm not sure where to go from here.






equivalence-relations set-partition




equivalence-relations set-partition






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 30 '18 at 3:05









Graham Kemp

84.7k43378




84.7k43378










asked Nov 30 '18 at 1:43









ClaireClaire

556




556








  • 1




    First, what is $[a]$?
    – Graham Kemp
    Nov 30 '18 at 1:49










  • $[a]$ is an equivalence class @GrahamKemp
    – Claire
    Nov 30 '18 at 1:50










  • For what equivalence relation?
    – Graham Kemp
    Nov 30 '18 at 1:51












  • I assume you are trying to show a step in the problem of showing that partitions are effectively equivalence relations and vice versa on a set. I further assume that you have some specific equivalence relation $sim$ for your specific partition $C$. You are likely intended to show that the equivalence class $[a]$ is equal to the part $U$. There is a big difference between "$[a]in U$" and "$[a]=U$."
    – JMoravitz
    Nov 30 '18 at 1:51








  • 1




    Yes I do, if you refer to this link and the question, it's this same question, this is the last part of the 4 part question. math.stackexchange.com/q/3019461/603668
    – Claire
    Nov 30 '18 at 1:53














  • 1




    First, what is $[a]$?
    – Graham Kemp
    Nov 30 '18 at 1:49










  • $[a]$ is an equivalence class @GrahamKemp
    – Claire
    Nov 30 '18 at 1:50










  • For what equivalence relation?
    – Graham Kemp
    Nov 30 '18 at 1:51












  • I assume you are trying to show a step in the problem of showing that partitions are effectively equivalence relations and vice versa on a set. I further assume that you have some specific equivalence relation $sim$ for your specific partition $C$. You are likely intended to show that the equivalence class $[a]$ is equal to the part $U$. There is a big difference between "$[a]in U$" and "$[a]=U$."
    – JMoravitz
    Nov 30 '18 at 1:51








  • 1




    Yes I do, if you refer to this link and the question, it's this same question, this is the last part of the 4 part question. math.stackexchange.com/q/3019461/603668
    – Claire
    Nov 30 '18 at 1:53








1




1




First, what is $[a]$?
– Graham Kemp
Nov 30 '18 at 1:49




First, what is $[a]$?
– Graham Kemp
Nov 30 '18 at 1:49












$[a]$ is an equivalence class @GrahamKemp
– Claire
Nov 30 '18 at 1:50




$[a]$ is an equivalence class @GrahamKemp
– Claire
Nov 30 '18 at 1:50












For what equivalence relation?
– Graham Kemp
Nov 30 '18 at 1:51






For what equivalence relation?
– Graham Kemp
Nov 30 '18 at 1:51














I assume you are trying to show a step in the problem of showing that partitions are effectively equivalence relations and vice versa on a set. I further assume that you have some specific equivalence relation $sim$ for your specific partition $C$. You are likely intended to show that the equivalence class $[a]$ is equal to the part $U$. There is a big difference between "$[a]in U$" and "$[a]=U$."
– JMoravitz
Nov 30 '18 at 1:51






I assume you are trying to show a step in the problem of showing that partitions are effectively equivalence relations and vice versa on a set. I further assume that you have some specific equivalence relation $sim$ for your specific partition $C$. You are likely intended to show that the equivalence class $[a]$ is equal to the part $U$. There is a big difference between "$[a]in U$" and "$[a]=U$."
– JMoravitz
Nov 30 '18 at 1:51






1




1




Yes I do, if you refer to this link and the question, it's this same question, this is the last part of the 4 part question. math.stackexchange.com/q/3019461/603668
– Claire
Nov 30 '18 at 1:53




Yes I do, if you refer to this link and the question, it's this same question, this is the last part of the 4 part question. math.stackexchange.com/q/3019461/603668
– Claire
Nov 30 '18 at 1:53










1 Answer
1






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oldest

votes


















2














This is a sequal to question Prove that $sim$ is an equivalence relation on the set $A$



In that you prove that $x,y$ belonging to the same partitioning set is an equivalence relation.



Here you are asked to prove if $ain$ the partitioning set $U$ that:



$[a] ={xin A| asim x} = U$.



As $a in Usubset A$ and that is distinct ($a$ is not in any other partitioning set) then



$[a] ={xin A| asim x}= {xin A| a, x in U} = {xin A|xin U} = {x in U} = U$.



That's all.






share|cite|improve this answer





















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    1 Answer
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    active

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    This is a sequal to question Prove that $sim$ is an equivalence relation on the set $A$



    In that you prove that $x,y$ belonging to the same partitioning set is an equivalence relation.



    Here you are asked to prove if $ain$ the partitioning set $U$ that:



    $[a] ={xin A| asim x} = U$.



    As $a in Usubset A$ and that is distinct ($a$ is not in any other partitioning set) then



    $[a] ={xin A| asim x}= {xin A| a, x in U} = {xin A|xin U} = {x in U} = U$.



    That's all.






    share|cite|improve this answer


























      2














      This is a sequal to question Prove that $sim$ is an equivalence relation on the set $A$



      In that you prove that $x,y$ belonging to the same partitioning set is an equivalence relation.



      Here you are asked to prove if $ain$ the partitioning set $U$ that:



      $[a] ={xin A| asim x} = U$.



      As $a in Usubset A$ and that is distinct ($a$ is not in any other partitioning set) then



      $[a] ={xin A| asim x}= {xin A| a, x in U} = {xin A|xin U} = {x in U} = U$.



      That's all.






      share|cite|improve this answer
























        2












        2








        2






        This is a sequal to question Prove that $sim$ is an equivalence relation on the set $A$



        In that you prove that $x,y$ belonging to the same partitioning set is an equivalence relation.



        Here you are asked to prove if $ain$ the partitioning set $U$ that:



        $[a] ={xin A| asim x} = U$.



        As $a in Usubset A$ and that is distinct ($a$ is not in any other partitioning set) then



        $[a] ={xin A| asim x}= {xin A| a, x in U} = {xin A|xin U} = {x in U} = U$.



        That's all.






        share|cite|improve this answer












        This is a sequal to question Prove that $sim$ is an equivalence relation on the set $A$



        In that you prove that $x,y$ belonging to the same partitioning set is an equivalence relation.



        Here you are asked to prove if $ain$ the partitioning set $U$ that:



        $[a] ={xin A| asim x} = U$.



        As $a in Usubset A$ and that is distinct ($a$ is not in any other partitioning set) then



        $[a] ={xin A| asim x}= {xin A| a, x in U} = {xin A|xin U} = {x in U} = U$.



        That's all.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 30 '18 at 2:37









        fleabloodfleablood

        68.7k22685




        68.7k22685






























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