Set theory Proof help

In summary, we are given an indexed family of sets {Ai| i \in I} and the index set I does not equal an empty set. We need to prove that the intersection of all sets in the family is an element of the power set of all sets in the family.
  • #1
13
0

Homework Statement


Suppose {Ai| i [tex]\in[/tex] I} is an indexed family of sets and I does
equal an empty set. Prove that [tex]\bigcap[/tex] i [tex]\in[/tex] I Ai
[tex]\in[/tex] [tex]\bigcap[/tex] i[tex]\in[/tex] I P(Ai ) and P(Ai) is the
power set of Ai

Homework Equations


none


The Attempt at a Solution


Suppose x [tex]\in[/tex] {Ai| i [tex]\in[/tex] I}. Let i be an arbitrary element of
I where x [tex]\in[/tex] Ai . Then let y be an arbitrary element of x. Since x
is an element of Ai and y [tex]\in[/tex] x it follows that ...

maybe i want to show that [tex]\bigcap[/tex] i [tex]\in[/tex] I Ai [tex]\subseteq[/tex] [tex]\bigcap[/tex] i [tex]\in[/tex] I Ai and then
I could say that [tex]\bigcap[/tex] i [tex]\in[/tex] I Ai [tex]\in[/tex] [tex]\bigcap[/tex] i[tex]\in[/tex] I P(Ai )
 
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  • #2
Let [tex]\left\{ A_{i} \right\}_{i \in I} [/tex] be your indexed set of family.

Do you mean this [tex]\bigcap_{i=1} A_i = \left\{ x : \forall i \in I: x \in A_i \right\} [/tex]?
 
  • #3
Yes sry about the horrible looking symbols
 
  • #4
nike5 said:

Homework Statement


Suppose {Ai| i [tex]\in[/tex] I} is an indexed family of sets and I does
equal an empty set.
Did you mean "does not equal and empty set"?
 
  • #5
I [tex]\neq[/tex] [tex]\oslash[/tex] is what I meant
 

1. What is set theory proof?

Set theory proof is a method of using logical reasoning and mathematical principles to demonstrate the validity of a statement or theorem within the framework of set theory. It involves defining sets, their elements, and operations on sets, and using axioms and rules of inference to prove the truth of a given proposition.

2. How do you construct a set theory proof?

To construct a set theory proof, you first need to clearly state the statement or theorem you are trying to prove. Then, you need to define all the relevant sets and operations involved. Next, use the axioms and rules of inference to make logical deductions and arrive at a conclusion. Finally, check that your proof is valid and complete, and that it follows all the necessary steps.

3. What are some common axioms used in set theory proofs?

Some common axioms used in set theory proofs include the axiom of extension, which states that two sets are equal if and only if they have the same elements, and the axiom of pairing, which allows for the creation of a set containing two specified elements. Other important axioms include the axiom of union, intersection, and power set.

4. Are there any strategies for approaching set theory proofs?

Yes, there are several strategies that can be helpful when approaching set theory proofs. These include breaking down the statement into simpler parts, using known theorems or lemmas, and working backwards from the conclusion. It can also be helpful to draw diagrams or use examples to gain a better understanding of the problem before attempting to prove it.

5. How can I check my set theory proof for errors?

To check your set theory proof for errors, it is important to carefully review each step and make sure it follows logically from the previous ones. You can also try to find counterexamples or cases where your proof does not hold. Additionally, it can be helpful to have someone else review your proof and provide feedback. If possible, use a computer program or proof assistant to verify the correctness of your proof.

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