# Thread: help with partition problems

1. ## 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.

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.

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### the set of positive integers greater than 100

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