# What are Positive Sets, Ultimate Sets?

• Dec 4th 2010, 02:47 AM
archon
What are Positive Sets, Ultimate Sets?
Could someone please explain what these are? I've got a general understanding of set theory. I've tried searching the web for a definition, but can't seem to find one.
• Dec 4th 2010, 08:00 AM
HallsofIvy
Those are not notions from "set theory" itself but from "numerical dynamics" where we have some kind of recursion formula, say \$\displaystyle a_n= f(a_{n-1})\$, or differential equation dy/dx= f(x,y). (The term "dynamics" is from the similarity to motion where the future position of a body is determined by its position now.)
If we are given some value, \$\displaystyle a_0\$ or \$\displaystyle y(x_0)\$ we could use those formulas to determine \$\displaystyle a_n\$ or y(x) for different values of n and x. In particular, the "positive set" is the set of all \$\displaystyle a_k\$ or \$\displaystyle y(x)\$ for k> 0 and x> 0. The "ultimate set" is the set of values for \$\displaystyle a_n\$ or \$\displaystyle y(x)\$ in the limit as n and x go to infinity.
• Dec 4th 2010, 08:21 AM
archon
Perhaps there are multiple definitions of positive and ultimate sets in mathematics, but within the context of what I know your definitions don't seem to work. Here are 2 questions from my assignment:

1. Given 4 sets A,B,C,D, express the following in terms of positive sets.

AcCc
ABCcD
ABcD
(BcCD)c
AcBcCcD
2. Express the following in terms of ultimate sets.

U - A - B + AB (here U is the universal set)
AB ∪ AC ∪ BC
AB + AC + BC
ABC ∪ ABD ∪ ACD ∪ BCD
ABC + ABD + ACD + BCD

Question 1 deals with cardinalities, In question 2 the examples with + or - deal with cardinalities. These are to be solved using inclusion/exclusion. In other words, N(AU B) = N(A) + N(B) - N(A Intersect B).
• Nov 26th 2011, 09:44 PM
Annatala
Re: What are Positive Sets, Ultimate Sets?
This is the second person to ask this question in as many days. Read your book/notes for the answer, or ask the instructor. This is not a common definition used in set theory and it is particular to your text.

(Also, tell me who your instructor is so I can inform her/him that multiple students are unable to complete the assignment due to lack of resources...) :P
• Nov 28th 2011, 02:47 AM
archon
Re: What are Positive Sets, Ultimate Sets?
You're a little late to the party. Check the post dates. If someone else is needing help on this topic I agree that they should ask their professor, because I could not find anything related to this on the whole of the internet(and it was not in my textbook either).
• Nov 28th 2011, 07:40 AM
Annatala
Re: What are Positive Sets, Ultimate Sets?
Oh wow. Why on Earth did this post show up at the top of my list then? I guess I probably clicked through from another post and didn't realize it.
• Nov 28th 2011, 03:36 PM
ehpoc
Re: What are Positive Sets, Ultimate Sets?
My assignment has the exact same question on positive sets.....judging by the post date, one year later.

You seem like the perfect person to put it in context for me. :D
• Nov 28th 2011, 04:39 PM
Plato
Re: What are Positive Sets, Ultimate Sets?
Quote:

Originally Posted by ehpoc
My assignment has the exact same question on positive sets.....judging by the post date, one year later.

These questions and notations are so extremely odd( read non-standard) to make one wonder from where they could possibly come.

Would you please share with us the source of these questions?
From what text material do these questions come?