Results 1 to 2 of 2

Thread: help with partition problems

  1. #1
    Member
    Joined
    Nov 2005
    Posts
    111

    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
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    12,028
    Thanks
    848
    Hello, braddy!

    A partition of a set $\displaystyle A$ divides $\displaystyle A$ 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 $\displaystyle A$ 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?

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


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

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

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

    It is a partition.



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

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

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

    It is not a partition.



    3) $\displaystyle P$ = the set of integers divisible by 3,
    . .$\displaystyle Q$ = the set of integers leaving a remainder of 1 when divided by 3,
    . .$\displaystyle R$ = 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 $\displaystyle P.$
    . . [2] the remainder is 1 . . . the integer is in $\displaystyle Q.$
    . . [3] the remainder is 2 . . . the integer is in $\displaystyle R.$

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

    It is a partition.



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

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

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

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

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

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

    It is a partition.

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. partition help
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Jun 28th 2010, 03:35 AM
  2. set theory-partition
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: Nov 22nd 2008, 02:14 PM
  3. Partition
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: May 1st 2008, 02:39 AM
  4. Partition help!!!
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: Mar 15th 2008, 08:57 PM
  5. Partition a Square
    Posted in the Geometry Forum
    Replies: 1
    Last Post: Apr 1st 2007, 07:43 AM

Search tags for this page

Search Tags


/mathhelpforum @mathhelpforum