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Math Help - Need help proving that two sets are equal

  1. #1
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    Need help proving that two sets are equal

    I need help with the following.
    Prove that:
    {3q : q ∈ Z} U {3q + 1 : q ∈ Z} U {3q + 2 : q ∈ Z} = Z
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  2. #2
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    Re: Need help proving that two sets are equal

    Use the fact that the remainder of every integer when divided by 3 is 0, 1 or 2.
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  3. #3
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    Re: Need help proving that two sets are equal

    In general you prove that " A= B", for A and B sets, by proving " A\subset B" and " B\subset A".

    And you prove " X\subset Y" by starting "if x\in X" and using the properties of X and Y to conclude "then x\in Y"

    Here, you have A= \{3q| q\in Z\}\cup \{3q+1| q\in Z\}\cup \{3q+ 2| q\in Z\} and B= Z.

    To show that A\subset Z is easy. is x\in A then
    i) x\in \{3q| q\in Z\}
    Since Z is "closed under multiplication" x= 3q, for any q in Z, is in Z
    or ii) x\in \{3q+ 1| q\in Z\}
    Since Z is "closed under multiplication" and "closed under addition", x= 3q+ 1, for any q in Z, is in Z.
    or iii) x\in \{3q+ 2} q\in Z\}
    Again, since Z is "closed under multiplication" and "closed under addition", x= 3q+ 2, for any q in Z, is in Z.

    Therefore A is a subset of Z.

    To prove the other way is only slightly harder. If x\in Z, then dividing x by 3 gives remainder 0, 1, or 2 since the remainder has to be a non-negative integer less than 3: x/3 "has quotient q with remainder r" so that x/3= q+ r/3 and x= 3q+ r.
    i) If the remainder is 0 then x= 3q where q is the quotient so x\in \{3q| q\in Z\}
    ii) If the remainder is 1 then x= 3q+ 1 where q is the quotient so x\in \{3q+ 1| q\in Z\}
    iii) If the remainder is 2 then x= 3q+ 2 where q is the quotient so x\in \{3q+ 2| q\in Z\}

    In any case, x\in A so Z\subset A

    Together those give A= \{3q| q\in Z\}\cup \{3q+1| q\in Z\}\cup \{3q+ 2| q\in Z\}= Z.
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