Discrete Math: prove an intersection from a given

In summary, to prove that B intersection A equals A, we use the given statement of A-B being an empty set. This means that all elements in A are also in B. Therefore, any element in A is also in B intersection A, proving that B intersection A equals A.
  • #1
JackRyan
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Discrete Math: prove B intersection A = A, given A-B = null set

1. Problem Statement:
Prove B [itex]\cap[/itex] A = A, given A-B = ∅ (empty set)

The Attempt at a Solution


xε(B[itex]\cap[/itex]A) => xεB and xεA => Logic given A-B = ∅ => xεA

I tried using A-B = A[itex]\cap[/itex]!B for xε(A[itex]\cap[/itex]!B)=∅ => xεA and x not in !B or x not in A and Xε!B

I am unsure how to fill in that logic section and prove that B[itex]\cap[/itex]A=A
 
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  • #2
The first direction is fine. If x is in A[itex]\bigcap[/itex]B then, certainly, x is in A (no need to use the given statement). Now suppose that x is in A and proceed by contradiction (to show that x is in A[itex]\bigcap[/itex]B). If x is not also in B determine what that implies about the given statement.
 

1. What is discrete math?

Discrete math is a branch of mathematics that deals with countable and finite sets of elements. It involves studying and analyzing mathematical structures such as graphs, networks, and logical systems.

2. What does it mean to prove an intersection in discrete math?

Proving an intersection in discrete math means using mathematical logic and reasoning to show that two or more sets have at least one element in common. This is typically done through a proof by contradiction or by using set notation and logical operations.

3. How do you prove an intersection from a given set?

To prove an intersection from a given set, you must first identify the sets involved and their elements. Then, you can use set notation and logical operations such as union, intersection, and complement to show that there is at least one common element between the sets.

4. Why is proving an intersection important in discrete math?

Proving an intersection is important in discrete math because it allows us to establish relationships between sets and understand their properties. It also helps us to solve problems and make logical deductions based on the elements in the sets.

5. Can you provide an example of proving an intersection in discrete math?

Yes, for example, let A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. To prove that these two sets have at least one common element, we can use set notation and write A ∩ B = {3, 4}. This shows that the sets have an intersection of {3, 4}, meaning they share the elements 3 and 4. This is a simple example, but in more complex problems, we may need to use proofs by contradiction or other methods to show the intersection.

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