Creating Five Distinct Partitioning Sets for A, Z, and R"

[chocolatelover]

Homework Statement

a. Let A={1,2,...10}. Describe a partition of A that gives rise to five distinct paritioning sets.
b.Describe a partition of Z that gives rise to five distinct partitioning sets
c. Describe a partition of R that gives rise to five distint partitioning sets

Homework Equations

The Attempt at a Solution

Could someone please explain to me how to describe a partion in general? Is this where you say that each level has a certain amount of multiples of the set?

Thank you very much

 

[HallsofIvy]

You "describe" a partition by telling what sets are in it! Are you clear on what a partition is? It is simply a collection of subsets of the original set such that each member of the original set is in one and only one of the subsets.

For example, if I were asked to find a partition of {1, 2, 3, 4, 5, 6, 7} consisting of "5 distinct sets" I might give {{1}, {2}, {3}, {4}, {5, 6, 7}}. That's a partition (each member of the set is in exactly one of those) and it has 5 distinct sets. That's all that's required.

Now, for "b.Describe a partition of Z that gives rise to five distinct partitioning sets", yes, one way to do that is to use "modulo 5"- each set containing all integers whose remainder, when divided by 5, is the same. Another perfectly valid answer, since it is not required that each set in a partition be the same size, would be {{all negative integers},{1},{2},{3}, {all integers larger than 3}}.

 

[chocolatelover]

Thank you very much

Part c. "Describe a partition of R that gives rise to fie distinct partitioning sets" could be the same thing as part b., right? For these types of problems, there isn't just one correct answer, right?

Thank you

 

[HallsofIvy]

No, (c) is not the same as (b). (b) asked for a partition of Z, the set of integers, so the partition must include only sets of integers. (c) is asking for a partition of R, the set of real numbers, so the partition must include all real numbers.

 

[chocolatelover]

Thank you very much

Regards