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cinar

  • 2 years ago

1) Prove that if A and B are countable, then \[A \cap B\] is also countable. 2) Prove A\(A\B)=B\(B\A)

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  1. sauravshakya
    • 2 years ago
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    A n B can never be greater than A and B ........ So 0<=A n B <=A and 0<=A n B <=B...... THus, if A and B are countable ......... A n B is also countable

  2. sauravshakya
    • 2 years ago
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    |dw:1346927959015:dw|

  3. RolyPoly
    • 2 years ago
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    @cinar Can we use Venn Diagrams to prove it? @sauravshakya I think for question 1 and 2, they are about the topic ''sets''.

  4. cinar
    • 2 years ago
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    @Rolypoly no we cant use venn diagrams

  5. helder_edwin
    • 2 years ago
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    \(\setminus\) is not division !!!!!!!!!!!!!!!!!!!!!!!!!!

  6. helder_edwin
    • 2 years ago
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    \[ \large A\setminus(B\setminus A)=A\cap(B\cap A^c)^c=A\cap(B^c\cup A)=A \] on the other hand \[ \large B\setminus(B\setminus A)=B\cap(B\cap A^c)^c=B\cap(B^c\cup A)= \] \[ \large = (B\cap B^c)\cup(B\cap A)=B\cap A \] they are not equal. unless \(A\subseteq B\).

  7. cinar
    • 2 years ago
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    \[A-(B-A)=A \cap B \]

  8. cinar
    • 2 years ago
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    \[B-(B-A)=A \cap B\]

  9. cinar
    • 2 years ago
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    this is a true statement, I just dont know how to prove it..

  10. helder_edwin
    • 2 years ago
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    Let \(A=\{a,b,c,d,e\}\) and \(B=\{a,i,u,e,o\}\) then \[ \large A\setminus(B\setminus A)=A\setminus\{i,u,o\}=A \] and \[ \large B\setminus(B\setminus A)=B\setminus\{i,u,o\}=\emptyset \]

  11. helder_edwin
    • 2 years ago
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    IT IS NOT TRUE !!!!!!!!!!!!!

  12. cinar
    • 2 years ago
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    sorry there is a typo the question is Prove A\(A\B)=B\(B\A)

  13. helder_edwin
    • 2 years ago
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    \[ \large A\setminus(A\setminus B)=A\cap(A\cap B^c)^c=A\cap(A^c\cup B) \] \[ \large =(A\cap A^c)\cup(A\cap B)=A\cap B \]

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