Hello, braddy!

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

We must see if the following is true:

. . (a) There is no "overlap" among the subsets.

. . (b)Allthe elements of are used.

You can answer these questions with someThinking.

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

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

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

. . (a) . . . There is no overlap

. . (b) . . . All of is used

Itisa partition.

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

. . (a) . . . They are disjoint

. . (b) . . . The is not included.

It isnota partition.

3) = the set of integers divisible by 3,

. . = the set of integers leaving a remainder of 1 when divided by 3,

. . = the set of integers leaving a remainder of 2 when divided by 3

You canthinkyour 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

. . [2] the remainder is 1 . . . the integer is in

. . [3] the remainder is 2 . . . the integer is in

. . (a) are disjoint.

. . (b)

Itisa partition.

4) = the set of integers less than -100,

. . = the set of integers with absolute value not exceeding 100,

. . = the set of integers greater than 100.

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

is easy: .

is easy: .

has integers , where . . . that is: .

. . Hence: .

. . (a) are disjoint.

. . (b) . . . all of is used

Itisa partition.