Prove $|A times A| geq |A|$











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For case 1, I proved that if $A$ is finite/countable, then $|Atimes A|$ is finite/countable. However, I don’t know how to proceed with showing $|Atimes A|geq|A|$. Specifically, I want to say for every $a_i$ in $A$, there is a corresponding $(a_i, a_k)$ in $|Atimes A|$ whose list is certainly greater than a single $a_i$ for a corresponding $i$. However, I was given a remark which states if $|A| geq |B|$, then there exists some $f:A to B$ that is onto. How would I show $f:|A times A|to|A|$ is onto?



Case 2 would be for $A$ is not countable.



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  • Please use the MathJax standard format and also wrtie down the problem and what you are doing or trying. As is, the problem will be closed shortly.
    – Will M.
    Nov 20 at 4:16










  • Hint: Assume $A$ is nonempty. Consider a specific $xin A$. Consider the set ${(a,x)~:~ain A}$ and how it compares to $A$ itself. Additional hint: $|Y|geq |X|$ if and only if there is some $Zsubseteq Y$ such that $|Z|=|X|$.
    – JMoravitz
    Nov 20 at 4:21

















up vote
-2
down vote

favorite












For case 1, I proved that if $A$ is finite/countable, then $|Atimes A|$ is finite/countable. However, I don’t know how to proceed with showing $|Atimes A|geq|A|$. Specifically, I want to say for every $a_i$ in $A$, there is a corresponding $(a_i, a_k)$ in $|Atimes A|$ whose list is certainly greater than a single $a_i$ for a corresponding $i$. However, I was given a remark which states if $|A| geq |B|$, then there exists some $f:A to B$ that is onto. How would I show $f:|A times A|to|A|$ is onto?



Case 2 would be for $A$ is not countable.



enter image description here










share|cite|improve this question
























  • Please use the MathJax standard format and also wrtie down the problem and what you are doing or trying. As is, the problem will be closed shortly.
    – Will M.
    Nov 20 at 4:16










  • Hint: Assume $A$ is nonempty. Consider a specific $xin A$. Consider the set ${(a,x)~:~ain A}$ and how it compares to $A$ itself. Additional hint: $|Y|geq |X|$ if and only if there is some $Zsubseteq Y$ such that $|Z|=|X|$.
    – JMoravitz
    Nov 20 at 4:21















up vote
-2
down vote

favorite









up vote
-2
down vote

favorite











For case 1, I proved that if $A$ is finite/countable, then $|Atimes A|$ is finite/countable. However, I don’t know how to proceed with showing $|Atimes A|geq|A|$. Specifically, I want to say for every $a_i$ in $A$, there is a corresponding $(a_i, a_k)$ in $|Atimes A|$ whose list is certainly greater than a single $a_i$ for a corresponding $i$. However, I was given a remark which states if $|A| geq |B|$, then there exists some $f:A to B$ that is onto. How would I show $f:|A times A|to|A|$ is onto?



Case 2 would be for $A$ is not countable.



enter image description here










share|cite|improve this question















For case 1, I proved that if $A$ is finite/countable, then $|Atimes A|$ is finite/countable. However, I don’t know how to proceed with showing $|Atimes A|geq|A|$. Specifically, I want to say for every $a_i$ in $A$, there is a corresponding $(a_i, a_k)$ in $|Atimes A|$ whose list is certainly greater than a single $a_i$ for a corresponding $i$. However, I was given a remark which states if $|A| geq |B|$, then there exists some $f:A to B$ that is onto. How would I show $f:|A times A|to|A|$ is onto?



Case 2 would be for $A$ is not countable.



enter image description here







discrete-mathematics proof-writing






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edited Nov 20 at 4:29

























asked Nov 20 at 4:14









darylnak

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156111












  • Please use the MathJax standard format and also wrtie down the problem and what you are doing or trying. As is, the problem will be closed shortly.
    – Will M.
    Nov 20 at 4:16










  • Hint: Assume $A$ is nonempty. Consider a specific $xin A$. Consider the set ${(a,x)~:~ain A}$ and how it compares to $A$ itself. Additional hint: $|Y|geq |X|$ if and only if there is some $Zsubseteq Y$ such that $|Z|=|X|$.
    – JMoravitz
    Nov 20 at 4:21




















  • Please use the MathJax standard format and also wrtie down the problem and what you are doing or trying. As is, the problem will be closed shortly.
    – Will M.
    Nov 20 at 4:16










  • Hint: Assume $A$ is nonempty. Consider a specific $xin A$. Consider the set ${(a,x)~:~ain A}$ and how it compares to $A$ itself. Additional hint: $|Y|geq |X|$ if and only if there is some $Zsubseteq Y$ such that $|Z|=|X|$.
    – JMoravitz
    Nov 20 at 4:21


















Please use the MathJax standard format and also wrtie down the problem and what you are doing or trying. As is, the problem will be closed shortly.
– Will M.
Nov 20 at 4:16




Please use the MathJax standard format and also wrtie down the problem and what you are doing or trying. As is, the problem will be closed shortly.
– Will M.
Nov 20 at 4:16












Hint: Assume $A$ is nonempty. Consider a specific $xin A$. Consider the set ${(a,x)~:~ain A}$ and how it compares to $A$ itself. Additional hint: $|Y|geq |X|$ if and only if there is some $Zsubseteq Y$ such that $|Z|=|X|$.
– JMoravitz
Nov 20 at 4:21






Hint: Assume $A$ is nonempty. Consider a specific $xin A$. Consider the set ${(a,x)~:~ain A}$ and how it compares to $A$ itself. Additional hint: $|Y|geq |X|$ if and only if there is some $Zsubseteq Y$ such that $|Z|=|X|$.
– JMoravitz
Nov 20 at 4:21












1 Answer
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To show that $f$ is onto, consider a sort of projection. For example, if



$$A = {1,2}$$



then



$$A times A = {(1,1);,;(1,2);,;(2,1);,;(2,2)}$$



We can define the function $f : Atimes A rightarrow A $ by



$$f(x,y)=x$$



Then $(1,1)$ and $(1,2)$ map to $1$, and $(2,1)$ and $(2,2)$ map to $2$. Thus, every element of $A$ has a preimage.



Can you see how this might prove $f$ is surjective in a more general case if $A$ is nonempty? (Or "onto," whichever terminology you prefer.) About the only nuance you would really need to consider for this would be $A = emptyset$, which you can handle separately and more trivially.






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    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    1
    down vote



    accepted










    To show that $f$ is onto, consider a sort of projection. For example, if



    $$A = {1,2}$$



    then



    $$A times A = {(1,1);,;(1,2);,;(2,1);,;(2,2)}$$



    We can define the function $f : Atimes A rightarrow A $ by



    $$f(x,y)=x$$



    Then $(1,1)$ and $(1,2)$ map to $1$, and $(2,1)$ and $(2,2)$ map to $2$. Thus, every element of $A$ has a preimage.



    Can you see how this might prove $f$ is surjective in a more general case if $A$ is nonempty? (Or "onto," whichever terminology you prefer.) About the only nuance you would really need to consider for this would be $A = emptyset$, which you can handle separately and more trivially.






    share|cite|improve this answer

























      up vote
      1
      down vote



      accepted










      To show that $f$ is onto, consider a sort of projection. For example, if



      $$A = {1,2}$$



      then



      $$A times A = {(1,1);,;(1,2);,;(2,1);,;(2,2)}$$



      We can define the function $f : Atimes A rightarrow A $ by



      $$f(x,y)=x$$



      Then $(1,1)$ and $(1,2)$ map to $1$, and $(2,1)$ and $(2,2)$ map to $2$. Thus, every element of $A$ has a preimage.



      Can you see how this might prove $f$ is surjective in a more general case if $A$ is nonempty? (Or "onto," whichever terminology you prefer.) About the only nuance you would really need to consider for this would be $A = emptyset$, which you can handle separately and more trivially.






      share|cite|improve this answer























        up vote
        1
        down vote



        accepted







        up vote
        1
        down vote



        accepted






        To show that $f$ is onto, consider a sort of projection. For example, if



        $$A = {1,2}$$



        then



        $$A times A = {(1,1);,;(1,2);,;(2,1);,;(2,2)}$$



        We can define the function $f : Atimes A rightarrow A $ by



        $$f(x,y)=x$$



        Then $(1,1)$ and $(1,2)$ map to $1$, and $(2,1)$ and $(2,2)$ map to $2$. Thus, every element of $A$ has a preimage.



        Can you see how this might prove $f$ is surjective in a more general case if $A$ is nonempty? (Or "onto," whichever terminology you prefer.) About the only nuance you would really need to consider for this would be $A = emptyset$, which you can handle separately and more trivially.






        share|cite|improve this answer












        To show that $f$ is onto, consider a sort of projection. For example, if



        $$A = {1,2}$$



        then



        $$A times A = {(1,1);,;(1,2);,;(2,1);,;(2,2)}$$



        We can define the function $f : Atimes A rightarrow A $ by



        $$f(x,y)=x$$



        Then $(1,1)$ and $(1,2)$ map to $1$, and $(2,1)$ and $(2,2)$ map to $2$. Thus, every element of $A$ has a preimage.



        Can you see how this might prove $f$ is surjective in a more general case if $A$ is nonempty? (Or "onto," whichever terminology you prefer.) About the only nuance you would really need to consider for this would be $A = emptyset$, which you can handle separately and more trivially.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 20 at 4:22









        Eevee Trainer

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