Nightmares with formal proofs in set theory

In summary, the conversation discusses the struggle of proving things in set theory and a specific homework problem involving the proof of a set equality related to domain. The speaker is seeking resources for help and the expert provides a general strategy for proving set equality.
  • #1
JasonJo
429
2
I am having a nightmare trying to prove things in set theory.

One of my homework problems is to prove that:

Dom(R U S) = Dom(R) U Dom(S)

but i have no idea how to really do this. my teacher never went over this stuff! IT'S SO AGGRAVATING!

can anyone reference a good site or book on how to prove things in set theory, such as the domain, inverse of function only if the function is one-to-one, etc?

ack!
 
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  • #2
The basic way to prove A is a subset of B is: Let x be a member of A. Then use what ever the definition of A is to show that x also satisfies the definition of B: x is a member of B.

Here, unfortunately, you haven't told us what R and S are and you haven't told us what "Dom" means. If R and S are functions I might guess that "Dom" means domain.
 
  • #3
R and S are relations, Dom means domain
 
  • #4
Suppose that (x,y) satisfies the relation RUS (i.e, (x,y) is in Dom(RUS)
Then, (x,y) either satisfies the relation R (i.e, (x,y) is in Dom(R)), or (x,y) satisfies the relation S (i.e, (x,y) is in Dom(S))
Thus, Dom(RUS) is contained within Dom(R)UDom(S).

I'll leave to you to show that Dom(R)UDom(S) is contained within Dom(RUS).
 

1. What is set theory and why is it important in formal proofs?

Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects. It provides a foundation for other areas of mathematics, as well as for computer science and philosophy. In formal proofs, set theory is important because it allows for precise definitions and logical reasoning to be used in proving mathematical statements.

2. What are formal proofs and how do they relate to set theory?

Formal proofs are a rigorous method of proving mathematical statements using a set of axioms and rules of inference. These proofs are based on the principles of logic and rely heavily on the concepts and notation of set theory, such as union, intersection, and subset.

3. Why do nightmares occur in formal proofs in set theory?

Nightmares in formal proofs can occur due to the complexity and abstract nature of set theory. This can make it challenging to understand and apply the rules of inference correctly, leading to errors in the proof. Additionally, the use of symbols and notation can also cause confusion or mistakes.

4. How can one avoid nightmares in formal proofs in set theory?

To avoid nightmares in formal proofs, it is important to have a strong understanding of set theory and the rules of inference. It is also helpful to practice and double check all steps of the proof to ensure accuracy. Seeking guidance from a mentor or instructor can also be beneficial in avoiding nightmares.

5. Can nightmares in formal proofs be beneficial?

While they can be frustrating, nightmares in formal proofs can also be beneficial as they challenge us to think critically and improve our problem-solving skills. They can also highlight areas of weakness in our understanding and provide opportunities for growth and learning.

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