Finding a counter-example to an alleged set identity

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Homework Help Overview

The discussion revolves around finding a counter-example to a set identity, specifically in the context of universal sets and their subsets. Participants are exploring the nature of universal sets and how they relate to the elements involved in the identity.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the use of finite versus infinite sets for counter-examples, with some suggesting that a finite universal set can suffice. Questions arise regarding the definition and implications of a universal set in mathematical contexts.

Discussion Status

The conversation is active, with participants providing insights into the nature of universal sets and suggesting practical approaches to constructing counter-examples. There is a recognition of differing interpretations of what constitutes a universal set.

Contextual Notes

Participants are navigating the distinction between "a" universal set and "the" universal set, indicating a need for clarity on definitions within the problem's framework. There is an emphasis on the theoretical context in which the sets A and B operate.

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Homework Statement



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Question #2.

Homework Equations


The Attempt at a Solution



I've drawn a venn diagram for the left-hand side and the right-hand side and I can see that they're not equal but how do I provide a counter-example for this? Wouldn't a counter-example require an infinite number of elements?
 
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You can make a counter example with a finite number of elements, in fact I made a counterexample with U containing only a couple elements. I'd recommend just making U a set with a couple elements and then try out a couple of subsets A and B until you get something that works. It shouldn't take particularly long.
 
kduna said:
I'd recommend just making U a set with a couple elements

You can do this? I thought U had to contain all the elements possible in Mathematics? Why is it called a universal set then?
 
The problem states "A" universal set. Not "the" universal set, which wouldn't really make sense.

http://mathworld.wolfram.com/UniversalSet.html

A set fixed within the framework of a theory and consisting of all objects considered in this theory.
 
ainster31 said:
You can do this? I thought U had to contain all the elements possible in Mathematics? Why is it called a universal set then?

You certainly can do this. The set is not called universal because it contains everything you could possibly think of mathematically. The term universal comes from the fact that we need to know that the sets A and B exist somewhere (in some universe) so that we have some backdrop to perform these set operations in. So we just say that A and B are subsets of some universal set U.

Try letting U = {1, 2, 3}.
 

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