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Set Theory Homework find ∪i=0Ai and ∩ i=0Ai

  1. Jul 11, 2011 #1
    1. The problem statement, all variables and given/known data

    Find ∪i=0Ai (with infinite symbol) and ∩ i=0Ai (with infinite symbol) in each of the cases when for each natural number
    i, Ai is defined as:

    1. Ai = {i,−i, i + 1,−(i + 1), i + 2,−(i + 2), . . .}
    2. Ai = {0, i, 2i}
    3. Ai = {x : x is a real number such that i < x < i + 1}
    4. Ai = {x : x is a real number strictly between − i and i}.

    2. Relevant equations



    3. The attempt at a solution

    I really have no clue on this part of set theory
     
  2. jcsd
  3. Jul 11, 2011 #2

    gb7nash

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    What it is that you're confused about? Do you know how to take a union of two sets? What about three sets? ...

    In any case, I'm assuming you need to find [itex]\bigcup_{i=0}^{\infty} A_{i}[/itex] for each problem. For these types of problems, just list out A0, A1, ...

    Let's look at 1.

    A0 = {0,1,-1,2,-2,...} (What is this set?)
    A1 = {1,-1,2,-2,...}
    .
    .
    .

    What happens if you take the union of these?
     
  4. Jul 11, 2011 #3

    SammyS

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    We help with homework. We don't do it for you.

    What do you know about any of this?

    What part of this problem do you need help with?

    For each part of the problem, ( 1., 2., 3., & 4.) write out the first few Ai. In other words, write out A0, A1, A2, A3, A4, A5.

    Check you textbook & notes for what is meant by: [itex]\displaystyle \bigcup_{1=0}^\infty \,A_i\,,\ \text{ and }\ \bigcap_{1=0}^\infty \,A_i\,.[/itex]

    Edited to include A0. DUH!!!
     
    Last edited: Jul 11, 2011
  5. Jul 11, 2011 #4
    ...ok i need to find [itex]\bigcup_{i=0}^{\infty} A_{i}[/itex] & the [itex]\bigcap_{i=0}^{\infty} A_{i}[/itex] for each

    i can see that A0 is a set of integers

    would the union be?

    A0 [itex]\bigcup_[/itex] A1 = {0,1,-1,2,-2,...}

    and the intersection?

    A0 [itex]\bigcap_[/itex] A1 = {1,-1,2,-2,...}

    is that it for question one then? lol i dont know if i have even done it
     
    Last edited: Jul 11, 2011
  6. Jul 11, 2011 #5

    gb7nash

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    Fix up your latex. For the union, the set is correct, but I have no idea what notation you're using. For taking the intersection, what elements do all sets have in common? Like Sammy suggested, write out A0, A1, A2, A3, ...
     
  7. Jul 11, 2011 #6
    ok
    1)
    A0 {0,1,-1,2,-2,...}
    A1 {1,-1,2,-2,...}
    A2 {-1,2,-2,...}
    A4 {2,-2,...}
    A5 {-2,...}
    think that right probably not...

    Would [itex]\bigcup_{i=0}^{\infty} A_{i}[/itex] simply be {0,1,-1,2,-2,...} ?
    would [itex]\bigcup_{i=0}^{\infty} A_{i}[/itex] be {-2,...}?
     
  8. Jul 11, 2011 #7

    gb7nash

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    A0 and A1 are correct. A2 is wrong. Look again at the problem and tell me what A2 is. What is A3? A4? etc..
     
  9. Jul 11, 2011 #8
    A2 {0,1,-1,2,-2,3,...}
    a3 {1,-1,2,-2,3...}
    a4 {0,1,-1,2,-2,3,-3,...}
    a5 {1,-1,2,-2,3,-3,...}
    ???
     
  10. Jul 11, 2011 #9

    gb7nash

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    Do you understand index notation? A2 means you plug in 2 wherever i is. For instance, A1 = {1,-1,1+1,-(1+1),...} = {1,-1,2,-2,...} Knowing this now, what's A2? A3? etc..
     
  11. Jul 11, 2011 #10
    as i said clueless ;D
    A2 {2,-2,3,-3,4,-4,...}
    A3 {3,-3,4,-4,5,-5,...}
    A4 {4,-4,5,-5,6,-6,...}
    A5 {5,-5,6,-6,7,-7,...}
     
  12. Jul 11, 2011 #11

    gb7nash

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    Those are correct now. What's the union and intersection?
     
  13. Jul 11, 2011 #12
    Would [itex]\bigcup_{i=0}^{\infty} A_{i}[/itex] simply be {0,1,-1,2,-2,3,-3,4,-4,5,-5,6,-6,7,-7,...} ?

    would [itex]\bigcap_{i=0}^{\infty} A_{i}[/itex] be {[itex]\o[/itex]}?
     
  14. Jul 11, 2011 #13
    for 2) Ai = {0, i, 2i}

    A0 {0,0,0}
    A1 {0,1,2}
    A2 {0,2,4}
    A3 {0,3,6}
    A4 {0,4,8}
    A5 {0,5,10}

    Would [itex]\bigcup_{i=0}^{\infty} A_{i}[/itex] simply be {0,1,2,3,4,5,6,8,10}

    would [itex]\bigcap_{i=0}^{\infty} A_{i}[/itex] be {0}?
     
  15. Jul 11, 2011 #14

    SammyS

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    For #1.
    Yes.
    If by {[itex]\o[/itex]} you mean the empty set, then yes.

    Empty set = {} = [itex]\emptyset[/itex] .
     
  16. Jul 11, 2011 #15

    SammyS

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    You forgot the "..." .
    Yes.
     
  17. Jul 11, 2011 #16
    thanks guys i should be ok now no doubt ill be back for help on something else
     
  18. Jul 11, 2011 #17
    .........ok now im stuck on 3) {x : x is a real number such that i < x < i + 1}
    how do you list the x?
    do you have to declare it?
    or do you just leave it as x i think im barking up the wrong tree but this is what i have got thus far...

    a0 = {0,x,1,...}
    a1 = {1,x,2,...}
    a2 = {2,x,3,...}
    a3 = {3,x,4,...}
    a4 = {4,x,5,...}
    a5 = {5,x,6,...} as i said i dont know what to do with the 'x'

    i get that its real numbers and i may have use a different technique problem is i dont know this technique
     
    Last edited: Jul 11, 2011
  19. Jul 11, 2011 #18

    SammyS

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    Do you know about open intervals on the number line ?

    If 0 < x < 1 , we write that as (0, 1) , etc.
     
  20. Jul 11, 2011 #19
    no i do not...

    so would a0 then be {(0, 1),...} etc
    then would the union be
    {(0,1),(1,2),(2,3),(3,4),(4,5),(5,6),...} or {0,1,2,3,4,5,6,...}
    i imagine the intersection would be [itex]\emptyset[/itex] whichever way
     
  21. Jul 11, 2011 #20

    SammyS

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    Actually, we write this as [itex](0,1)\cup(1,2)\cup(2,3)\cup(3,4)\cup(4,5)\cup(5,6)\dots[/itex] An alternative way to write this is, { x | x > 0 and x ≠ 1, 2, 3, 4, ...}.
    In fact, none of the integers, {0,1,2,3,4,5,6,...} are in the sets A0 , A1 , A1 , ... nor are they in the union of the Ai .

    Yes, you're right here.
     
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