## cinar 3 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)

1. sauravshakya

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

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3. RolyPoly

@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

@Rolypoly no we cant use venn diagrams

5. helder_edwin

\(\setminus\) is not division !!!!!!!!!!!!!!!!!!!!!!!!!!

6. helder_edwin

\[ \large A\setminus(B\setminus A)=A\cap(B\cap A^c)^c=A\cap(B^c\cup A)=A \] 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

\[A-(B-A)=A \cap B \]

8. cinar

\[B-(B-A)=A \cap B\]

9. cinar

this is a true statement, I just dont know how to prove it..

10. helder_edwin

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

IT IS NOT TRUE !!!!!!!!!!!!!

12. cinar

sorry there is a typo the question is Prove A\(A\B)=B\(B\A)

13. helder_edwin

\[ \large A\setminus(A\setminus B)=A\cap(A\cap B^c)^c=A\cap(A^c\cup B) \] \[ \large =(A\cap A^c)\cup(A\cap B)=A\cap B \]