Set Theory Proof Help: Proving C∩D=Ø When A⊆C and B⊆D

In summary, the conversation is about a user seeking help with a set theory problem involving sets A, B, C, and D. The problem states that if A is a subset of C and B is a subset of D, and the intersection of A and B is empty, then the intersection of C and D must also be empty. The user also clarifies that this is a biconditional and asks for help in proving it. The conversation is then closed as the question was posted in the wrong forum.
  • #1
dainty77
9
0
Hey guys!

I am new to this forum but saw the helpful posts on set theory proofs and wondered if I could finally get some help with this problem:

Suppose A, B, C, and D are sets with A⊆C and B⊆D. If A∩B=Ø then C∩D=Ø.

This is a biconditional so I have to prove it both ways correct?

Any help would be greatly appreciated!
 
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  • #2
This belongs in the homework forum. Do you have some attempt?

I don't see a biconditional in your statement.
 
  • #3
Oh my mistake!
 
  • #4
Thread closed as dainty77 posted this question (per R136a1's suggestion) in the homework section.
 
  • #5


Hi there!

Yes, you are correct. In order to prove a biconditional statement, you need to prove it in both directions. So in this case, you need to prove that if A∩B=Ø, then C∩D=Ø and also that if C∩D=Ø, then A∩B=Ø.

To prove the first direction, assume that A∩B=Ø. We want to show that C∩D=Ø. Since A⊆C and B⊆D, every element in A is also in C and every element in B is also in D. Therefore, if A∩B=Ø, then there cannot be any elements in C∩D, since all elements in C∩D must be in both C and D. This means that C∩D=Ø, as desired.

To prove the second direction, assume that C∩D=Ø. We want to show that A∩B=Ø. Since A⊆C and B⊆D, every element in A is also in C and every element in B is also in D. Therefore, if C∩D=Ø, then there cannot be any elements in A∩B, since all elements in A∩B must be in both A and B. This means that A∩B=Ø, as desired.

I hope this helps! Let me know if you have any other questions. Good luck with your proof!
 

Related to Set Theory Proof Help: Proving C∩D=Ø When A⊆C and B⊆D

1. What is set theory?

Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects. It is used to formalize and analyze mathematical concepts and structures, and to provide a foundation for other areas of mathematics.

2. What is a set theory proof?

A set theory proof is a logical argument that uses the basic principles and axioms of set theory to demonstrate the truth of a mathematical statement. It involves constructing a series of steps, using logical reasoning, to show that the statement is true.

3. How do I approach a set theory proof?

To approach a set theory proof, it is important to first understand the definitions and axioms of set theory. Then, carefully read and analyze the statement to be proved, and determine what assumptions or principles can be used to support the proof. Construct a logical argument using these principles to demonstrate the truth of the statement.

4. What are some common strategies for solving set theory proofs?

Some common strategies for solving set theory proofs include using mathematical induction, proof by contradiction, and direct proof. It is also important to break the proof down into smaller, more manageable steps, and to use diagrams or visual representations to aid in understanding and solving the problem.

5. Are there any resources available to help with set theory proofs?

Yes, there are many resources available to help with set theory proofs, including textbooks, online tutorials and courses, and mathematical forums and communities. It can also be helpful to work with a study group or seek assistance from a math tutor or teacher.

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