Prove $Acap(A'cap B')'=A$ using set equivalence laws

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











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2
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I am having trouble using set equivalence laws to prove the following.



$$Acap(A'cap B')'=A$$



Any help would be greatly appreciated.




elementary-set-theory




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  • use cap cup to write intersection and union in latex
    – Deepesh Meena
    Aug 25 at 2:01














up vote
2
down vote

favorite












I am having trouble using set equivalence laws to prove the following.



$$Acap(A'cap B')'=A$$



Any help would be greatly appreciated.




elementary-set-theory




share|cite|improve this question






















  • use cap cup to write intersection and union in latex
    – Deepesh Meena
    Aug 25 at 2:01












up vote
2
down vote

favorite









up vote
2
down vote

favorite











I am having trouble using set equivalence laws to prove the following.



$$Acap(A'cap B')'=A$$



Any help would be greatly appreciated.




elementary-set-theory




share|cite|improve this question














I am having trouble using set equivalence laws to prove the following.



$$Acap(A'cap B')'=A$$



Any help would be greatly appreciated.





elementary-set-theory





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 25 at 2:10









zzuussee

2,364524




2,364524










asked Aug 25 at 1:44









Goldmuskhex

233




233











  • use cap cup to write intersection and union in latex
    – Deepesh Meena
    Aug 25 at 2:01
















  • use cap cup to write intersection and union in latex
    – Deepesh Meena
    Aug 25 at 2:01















use cap cup to write intersection and union in latex
– Deepesh Meena
Aug 25 at 2:01




use cap cup to write intersection and union in latex
– Deepesh Meena
Aug 25 at 2:01










3 Answers
3






active

oldest

votes

















up vote
0
down vote



accepted










I'll give the algebraic derivation of the statement in supplement to Davids nice pictorials.



By the De Morgan laws for the complement, we first derive



$$Acap (A'cap B')'=Acap ((A')'cup (B')')$$



By elementary properties of the complement, that is $(X')'=X$, we have



$$Acap ((A')'cup (B')')=Acap (Acup B)$$



Now, $cap$ distributes over $cup$, i.e.



$$Acap (Acup B)=(Acap A)cup (Acap B)=Acup (Acap B)$$



as $Acap A=A$. Now, as $Acap Bsubseteq A$, we have $Acup (Acap B)=A$.






share|cite|improve this answer
















  • 1




    I would also like to know the reason for the downvote.
    – zzuussee
    Aug 25 at 2:07










  • I think the possible reason for downvote is that you gave a whole answer for a PSQ where the OP has showed no effort.
    – user572932
    Aug 25 at 2:41






  • 2




    I certainly didn't downvote your answer. It was very clear and I very much appreciate the time and effort that you took to answer it for me.
    – Goldmuskhex
    Aug 25 at 4:02










  • Thank you very much, I appreciate it. Glad I could help.
    – zzuussee
    Aug 25 at 10:26

















up vote
2
down vote













Sometimes figures are worth 1000 words:



enter image description here



enter image description here



enter image description here



enter image description here






share|cite|improve this answer



























    up vote
    1
    down vote













    $$A cap (A'cap B')'$$



    as by morgan's law $(P cap Q)'= P'cup Q' $



    thus $$(A'cap B')'=A cup B$$



    $$A cap (A cup B) $$
    $$(A cap A) cup (A cap B)$$
    $$A cap A =A $$
    $$A cup (A cap B)$$
    By Venn diagram, it is clear that it is nothing but A






    share|cite|improve this answer
















    • 1




      why the downvote though I don't understand??
      – Deepesh Meena
      Aug 25 at 2:02










    • I think the possible reason for downvote is that you gave a whole answer for a PSQ where the OP has showed no effort.
      – user572932
      Aug 25 at 2:42






    • 1




      but this is just crazy on this site if I write a hint I get downvoted or if I write a full answer I get downvoted don't know what to do just pray
      – Deepesh Meena
      Aug 25 at 2:44






    • 1




      I certainly didn't downvote your answer. It was very clear and I very much appreciate the time and effort that you took to answer it for me.
      – Goldmuskhex
      Aug 25 at 4:02










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    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    0
    down vote



    accepted










    I'll give the algebraic derivation of the statement in supplement to Davids nice pictorials.



    By the De Morgan laws for the complement, we first derive



    $$Acap (A'cap B')'=Acap ((A')'cup (B')')$$



    By elementary properties of the complement, that is $(X')'=X$, we have



    $$Acap ((A')'cup (B')')=Acap (Acup B)$$



    Now, $cap$ distributes over $cup$, i.e.



    $$Acap (Acup B)=(Acap A)cup (Acap B)=Acup (Acap B)$$



    as $Acap A=A$. Now, as $Acap Bsubseteq A$, we have $Acup (Acap B)=A$.






    share|cite|improve this answer
















    • 1




      I would also like to know the reason for the downvote.
      – zzuussee
      Aug 25 at 2:07










    • I think the possible reason for downvote is that you gave a whole answer for a PSQ where the OP has showed no effort.
      – user572932
      Aug 25 at 2:41






    • 2




      I certainly didn't downvote your answer. It was very clear and I very much appreciate the time and effort that you took to answer it for me.
      – Goldmuskhex
      Aug 25 at 4:02










    • Thank you very much, I appreciate it. Glad I could help.
      – zzuussee
      Aug 25 at 10:26














    up vote
    0
    down vote



    accepted










    I'll give the algebraic derivation of the statement in supplement to Davids nice pictorials.



    By the De Morgan laws for the complement, we first derive



    $$Acap (A'cap B')'=Acap ((A')'cup (B')')$$



    By elementary properties of the complement, that is $(X')'=X$, we have



    $$Acap ((A')'cup (B')')=Acap (Acup B)$$



    Now, $cap$ distributes over $cup$, i.e.



    $$Acap (Acup B)=(Acap A)cup (Acap B)=Acup (Acap B)$$



    as $Acap A=A$. Now, as $Acap Bsubseteq A$, we have $Acup (Acap B)=A$.






    share|cite|improve this answer
















    • 1




      I would also like to know the reason for the downvote.
      – zzuussee
      Aug 25 at 2:07










    • I think the possible reason for downvote is that you gave a whole answer for a PSQ where the OP has showed no effort.
      – user572932
      Aug 25 at 2:41






    • 2




      I certainly didn't downvote your answer. It was very clear and I very much appreciate the time and effort that you took to answer it for me.
      – Goldmuskhex
      Aug 25 at 4:02










    • Thank you very much, I appreciate it. Glad I could help.
      – zzuussee
      Aug 25 at 10:26












    up vote
    0
    down vote



    accepted







    up vote
    0
    down vote



    accepted






    I'll give the algebraic derivation of the statement in supplement to Davids nice pictorials.



    By the De Morgan laws for the complement, we first derive



    $$Acap (A'cap B')'=Acap ((A')'cup (B')')$$



    By elementary properties of the complement, that is $(X')'=X$, we have



    $$Acap ((A')'cup (B')')=Acap (Acup B)$$



    Now, $cap$ distributes over $cup$, i.e.



    $$Acap (Acup B)=(Acap A)cup (Acap B)=Acup (Acap B)$$



    as $Acap A=A$. Now, as $Acap Bsubseteq A$, we have $Acup (Acap B)=A$.






    share|cite|improve this answer












    I'll give the algebraic derivation of the statement in supplement to Davids nice pictorials.



    By the De Morgan laws for the complement, we first derive



    $$Acap (A'cap B')'=Acap ((A')'cup (B')')$$



    By elementary properties of the complement, that is $(X')'=X$, we have



    $$Acap ((A')'cup (B')')=Acap (Acup B)$$



    Now, $cap$ distributes over $cup$, i.e.



    $$Acap (Acup B)=(Acap A)cup (Acap B)=Acup (Acap B)$$



    as $Acap A=A$. Now, as $Acap Bsubseteq A$, we have $Acup (Acap B)=A$.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Aug 25 at 1:55









    zzuussee

    2,364524




    2,364524







    • 1




      I would also like to know the reason for the downvote.
      – zzuussee
      Aug 25 at 2:07










    • I think the possible reason for downvote is that you gave a whole answer for a PSQ where the OP has showed no effort.
      – user572932
      Aug 25 at 2:41






    • 2




      I certainly didn't downvote your answer. It was very clear and I very much appreciate the time and effort that you took to answer it for me.
      – Goldmuskhex
      Aug 25 at 4:02










    • Thank you very much, I appreciate it. Glad I could help.
      – zzuussee
      Aug 25 at 10:26












    • 1




      I would also like to know the reason for the downvote.
      – zzuussee
      Aug 25 at 2:07










    • I think the possible reason for downvote is that you gave a whole answer for a PSQ where the OP has showed no effort.
      – user572932
      Aug 25 at 2:41






    • 2




      I certainly didn't downvote your answer. It was very clear and I very much appreciate the time and effort that you took to answer it for me.
      – Goldmuskhex
      Aug 25 at 4:02










    • Thank you very much, I appreciate it. Glad I could help.
      – zzuussee
      Aug 25 at 10:26







    1




    1




    I would also like to know the reason for the downvote.
    – zzuussee
    Aug 25 at 2:07




    I would also like to know the reason for the downvote.
    – zzuussee
    Aug 25 at 2:07












    I think the possible reason for downvote is that you gave a whole answer for a PSQ where the OP has showed no effort.
    – user572932
    Aug 25 at 2:41




    I think the possible reason for downvote is that you gave a whole answer for a PSQ where the OP has showed no effort.
    – user572932
    Aug 25 at 2:41




    2




    2




    I certainly didn't downvote your answer. It was very clear and I very much appreciate the time and effort that you took to answer it for me.
    – Goldmuskhex
    Aug 25 at 4:02




    I certainly didn't downvote your answer. It was very clear and I very much appreciate the time and effort that you took to answer it for me.
    – Goldmuskhex
    Aug 25 at 4:02












    Thank you very much, I appreciate it. Glad I could help.
    – zzuussee
    Aug 25 at 10:26




    Thank you very much, I appreciate it. Glad I could help.
    – zzuussee
    Aug 25 at 10:26










    up vote
    2
    down vote













    Sometimes figures are worth 1000 words:



    enter image description here



    enter image description here



    enter image description here



    enter image description here






    share|cite|improve this answer
























      up vote
      2
      down vote













      Sometimes figures are worth 1000 words:



      enter image description here



      enter image description here



      enter image description here



      enter image description here






      share|cite|improve this answer






















        up vote
        2
        down vote










        up vote
        2
        down vote









        Sometimes figures are worth 1000 words:



        enter image description here



        enter image description here



        enter image description here



        enter image description here






        share|cite|improve this answer












        Sometimes figures are worth 1000 words:



        enter image description here



        enter image description here



        enter image description here



        enter image description here







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Aug 25 at 1:51









        David G. Stork

        8,05621232




        8,05621232




















            up vote
            1
            down vote













            $$A cap (A'cap B')'$$



            as by morgan's law $(P cap Q)'= P'cup Q' $



            thus $$(A'cap B')'=A cup B$$



            $$A cap (A cup B) $$
            $$(A cap A) cup (A cap B)$$
            $$A cap A =A $$
            $$A cup (A cap B)$$
            By Venn diagram, it is clear that it is nothing but A






            share|cite|improve this answer
















            • 1




              why the downvote though I don't understand??
              – Deepesh Meena
              Aug 25 at 2:02










            • I think the possible reason for downvote is that you gave a whole answer for a PSQ where the OP has showed no effort.
              – user572932
              Aug 25 at 2:42






            • 1




              but this is just crazy on this site if I write a hint I get downvoted or if I write a full answer I get downvoted don't know what to do just pray
              – Deepesh Meena
              Aug 25 at 2:44






            • 1




              I certainly didn't downvote your answer. It was very clear and I very much appreciate the time and effort that you took to answer it for me.
              – Goldmuskhex
              Aug 25 at 4:02














            up vote
            1
            down vote













            $$A cap (A'cap B')'$$



            as by morgan's law $(P cap Q)'= P'cup Q' $



            thus $$(A'cap B')'=A cup B$$



            $$A cap (A cup B) $$
            $$(A cap A) cup (A cap B)$$
            $$A cap A =A $$
            $$A cup (A cap B)$$
            By Venn diagram, it is clear that it is nothing but A






            share|cite|improve this answer
















            • 1




              why the downvote though I don't understand??
              – Deepesh Meena
              Aug 25 at 2:02










            • I think the possible reason for downvote is that you gave a whole answer for a PSQ where the OP has showed no effort.
              – user572932
              Aug 25 at 2:42






            • 1




              but this is just crazy on this site if I write a hint I get downvoted or if I write a full answer I get downvoted don't know what to do just pray
              – Deepesh Meena
              Aug 25 at 2:44






            • 1




              I certainly didn't downvote your answer. It was very clear and I very much appreciate the time and effort that you took to answer it for me.
              – Goldmuskhex
              Aug 25 at 4:02












            up vote
            1
            down vote










            up vote
            1
            down vote









            $$A cap (A'cap B')'$$



            as by morgan's law $(P cap Q)'= P'cup Q' $



            thus $$(A'cap B')'=A cup B$$



            $$A cap (A cup B) $$
            $$(A cap A) cup (A cap B)$$
            $$A cap A =A $$
            $$A cup (A cap B)$$
            By Venn diagram, it is clear that it is nothing but A






            share|cite|improve this answer












            $$A cap (A'cap B')'$$



            as by morgan's law $(P cap Q)'= P'cup Q' $



            thus $$(A'cap B')'=A cup B$$



            $$A cap (A cup B) $$
            $$(A cap A) cup (A cap B)$$
            $$A cap A =A $$
            $$A cup (A cap B)$$
            By Venn diagram, it is clear that it is nothing but A







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Aug 25 at 1:56









            Deepesh Meena

            2,766721




            2,766721







            • 1




              why the downvote though I don't understand??
              – Deepesh Meena
              Aug 25 at 2:02










            • I think the possible reason for downvote is that you gave a whole answer for a PSQ where the OP has showed no effort.
              – user572932
              Aug 25 at 2:42






            • 1




              but this is just crazy on this site if I write a hint I get downvoted or if I write a full answer I get downvoted don't know what to do just pray
              – Deepesh Meena
              Aug 25 at 2:44






            • 1




              I certainly didn't downvote your answer. It was very clear and I very much appreciate the time and effort that you took to answer it for me.
              – Goldmuskhex
              Aug 25 at 4:02












            • 1




              why the downvote though I don't understand??
              – Deepesh Meena
              Aug 25 at 2:02










            • I think the possible reason for downvote is that you gave a whole answer for a PSQ where the OP has showed no effort.
              – user572932
              Aug 25 at 2:42






            • 1




              but this is just crazy on this site if I write a hint I get downvoted or if I write a full answer I get downvoted don't know what to do just pray
              – Deepesh Meena
              Aug 25 at 2:44






            • 1




              I certainly didn't downvote your answer. It was very clear and I very much appreciate the time and effort that you took to answer it for me.
              – Goldmuskhex
              Aug 25 at 4:02







            1




            1




            why the downvote though I don't understand??
            – Deepesh Meena
            Aug 25 at 2:02




            why the downvote though I don't understand??
            – Deepesh Meena
            Aug 25 at 2:02












            I think the possible reason for downvote is that you gave a whole answer for a PSQ where the OP has showed no effort.
            – user572932
            Aug 25 at 2:42




            I think the possible reason for downvote is that you gave a whole answer for a PSQ where the OP has showed no effort.
            – user572932
            Aug 25 at 2:42




            1




            1




            but this is just crazy on this site if I write a hint I get downvoted or if I write a full answer I get downvoted don't know what to do just pray
            – Deepesh Meena
            Aug 25 at 2:44




            but this is just crazy on this site if I write a hint I get downvoted or if I write a full answer I get downvoted don't know what to do just pray
            – Deepesh Meena
            Aug 25 at 2:44




            1




            1




            I certainly didn't downvote your answer. It was very clear and I very much appreciate the time and effort that you took to answer it for me.
            – Goldmuskhex
            Aug 25 at 4:02




            I certainly didn't downvote your answer. It was very clear and I very much appreciate the time and effort that you took to answer it for me.
            – Goldmuskhex
            Aug 25 at 4:02

















             

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