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Math Help - help with partition problems

  1. #1
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    help with partition problems

    The context is Discrete math /relation

    Hi I need help with this problem I have some trouble with partitions:

    Which of these collections of subsets are partitions of the set of integers?

    1- The set of even integer and the set of odd integers.

    2- the set of positive integer and the set of negative integers.

    3- the set of integers divisible by 3, the set of integers leaving a remainder of 1 when divided by 3, and the set of integers divisible by 3, the set of integers leaving a remainder of 2 when divided by 3.

    4- The set of integers less than -100, the set of integers with absolute value not exceeding 100, and the set of integers greater than 100.

    5- the set of integers not divisible by 3, the set of even integers and the set of intger that leave a remainder of 3 when divided by 6.


    Thank you for your help
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  2. #2
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    Hello, braddy!

    A partition of a set A divides A into a number of disjoint subsets.

    We must see if the following is true:
    . . (a) There is no "overlap" among the subsets.
    . . (b) All the elements of A are used.

    You can answer these questions with some Thinking.

    I'll baby-talk through most of them . . .


    Which of these collections of subsets are partitions of the set of integers?

    I \;= \;\{\hdots\,\text{-}3,\,\text{-}2,\,\text{-}1,\,0,\,1,\,2,\,3,\,\hdots\}


    1) A = the set of even integers, B = the set of odd integers

    A \:=\:\{\hdots\,\text{-}5,\,\text{-}3,\,\text{-}1,\,1,\,3,\,5,\,\hdots\}
    B \:=\:\{\hdots,\,\text{-}6,\,\text{-}4,\,\text{-}2,\,0,\,2,\,4,\,6,\,\hdots\}

    . . (a) A \cap B \:=\:\emptyset . . . There is no overlap
    . . (b) A \cup B \:=\:I . . . All of I is used

    It is a partition.



    2) C = the set of positive integers, D = the set of negative integers

    C\:= \:\{1,\,2,\,3,\,4,\,\hdots\}
    D\:= \:\{\text{-}1,\,\text{-}2,\,\text{-}3,\,\text{-}4,\,\hdots\}

    . . (a) C \cap D\:=\:\emptyset . . . They are disjoint
    . . (b) C \cup D \:\neq\:I . . . The 0 is not included.

    It is not a partition.



    3) P = the set of integers divisible by 3,
    . . Q = the set of integers leaving a remainder of 1 when divided by 3,
    . . R = the set of integers leaving a remainder of 2 when divided by 3

    You can think your way through this one.

    When we divide an integer by 3, only three things can happen:
    . . [1] the remainder is 0 . . . the integer is in P.
    . . [2] the remainder is 1 . . . the integer is in Q.
    . . [3] the remainder is 2 . . . the integer is in R.

    . . (a) P,\,Q,\,R are disjoint.
    . . (b) P \cup Q \cup R \:=\:I

    It is a partition.



    4) A = the set of integers less than -100,
    . . B = the set of integers with absolute value not exceeding 100,
    . . C = the set of integers greater than 100.

    This one sounds complicated, but let's take baby steps . . .

    A is easy: . A \:=\:\{\hdots\,\text{-}104,\,\text{-}103,\,\text{-}102,\,\text{-}101\}

    C is easy: . C\:=\:\{101,\,102,\,103,\,104,\,\hdots\}

    B has integers n, where |n| \leq 100 . . . that is: . -100 \leq n \leq 100
    . . Hence: . B \:=\:\{\text{-}100,\,\text{-}99,\,\text{-}98,\,\hdots\,98,\,99,\,100\}

    . . (a) A,\,B,\,C are disjoint.
    . . (b) A \cup B \cup C \:=\:I . . . all of I is used

    It is a partition.

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