Proof of the identity A\(A\B)=B

In summary, the conversation was about trying to prove the identity A \ (A \ B) = B from Munkres' Topology. The process involved using the definition of A \ B and manipulating the sets until reaching the conclusion that the identity was indeed true. However, it was pointed out that the identity cannot hold for all A and B, and it is necessary to know the overall inclusion relation between the two sets.
  • #1
VladZH
56
1
I'm trying to proof an identity from Munkres' Topology

A \ ( A \ B ) = B

By definition A \ B = {x : x in A and x not in B}

A \( A \ B) = A \ (A ∩ Bc) = A ∩ (A ∩ Bc)c = A ∩ (Ac ∪ B) = (A ∩ Ac) ∪ (A ∩ B) = ∅ ∪ (A ∩ B) = A ∩ B

What did I miss?
 
Physics news on Phys.org
  • #2
VladZH said:
I'm trying to proof an identity from Munkres' Topology

A \ ( A \ B ) = B

By definition A \ B = {x : x in A and x not in B}

A \( A \ B) = A \ (A ∩ Bc) = A ∩ (A ∩ Bc)c = A ∩ (Ac ∪ B) = (A ∩ Ac) ∪ (A ∩ B) = ∅ ∪ (A ∩ B) = A ∩ B

What did I miss?
Either that ##A=X## is the entire space, or you've found a typo. Just consider a point ##b\in B\text{ \ }A##. It is clearly in ##B## but never in any set ##A\text{ \ }C## whatever ##C## might be; except ##A=X## of course.
 
  • Like
Likes VladZH
  • #3
VladZH said:
I'm trying to proof an identity from Munkres' Topology

A \ ( A \ B ) = B

By definition A \ B = {x : x in A and x not in B}

A \( A \ B) = A \ (A ∩ Bc) = A ∩ (A ∩ Bc)c = A ∩ (Ac ∪ B) = (A ∩ Ac) ∪ (A ∩ B) = ∅ ∪ (A ∩ B) = A ∩ B

What did I miss?

Perhaps even more simply, from the definition it is clear that ##A \text{ \ }X \subset A##. So, the identity as given cannot hold for all ##A, B##.
 
  • Like
Likes VladZH
  • #4
VladZH said:
I'm trying to proof an identity from Munkres' Topology

A \ ( A \ B ) = B

By definition A \ B = {x : x in A and x not in B}

A \( A \ B) = A \ (A ∩ Bc) = A ∩ (A ∩ Bc)c = A ∩ (Ac ∪ B) = (A ∩ Ac) ∪ (A ∩ B) = ∅ ∪ (A ∩ B) = A ∩ B

What did I miss?

You missed nothing. This is correct.
 
  • Like
Likes VladZH
  • #5
Thank you, guys. Seems like I confused with the formultaion
 
  • #6
You can always resort to brute force by trying to show every element of B is a subset of A\(A\B) and viceversa. But, yes, you need to know the overall inclusion relation between A and B.
 

What is the proof of the identity A(A\B)=B?

The proof of the identity A(A\B)=B is a mathematical concept that shows the equality of two expressions, A(A\B) and B, for all possible values of the variables involved. It is a fundamental concept in algebra and is used to simplify and solve equations.

What is the purpose of proving the identity A(A\B)=B?

The purpose of proving this identity is to show that the two expressions, A(A\B) and B, are equivalent and can be used interchangeably in mathematical equations. This can help simplify complex equations and make them easier to solve.

What is the process for proving the identity A(A\B)=B?

The process for proving this identity involves manipulating the expressions using algebraic properties and rules, such as the distributive property and the associative property, until they are simplified and equal to each other. This process may involve several steps and may vary depending on the specific expressions involved.

What are some real-world applications of the identity A(A\B)=B?

This identity has many real-world applications, such as in physics, engineering, and economics. For example, it can be used to calculate the force of an object in motion or to determine the cost of a product with a discount. It is also commonly used in computer programming and data analysis.

What are some common mistakes when proving the identity A(A\B)=B?

Some common mistakes when proving this identity include forgetting to distribute the A in the expression, misapplying algebraic rules, and making careless errors in calculations. It is important to pay attention to detail and double-check the steps taken to ensure the correct result is obtained.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
216
  • Set Theory, Logic, Probability, Statistics
Replies
0
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
60
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
731
  • Set Theory, Logic, Probability, Statistics
Replies
11
Views
479
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
963
  • Set Theory, Logic, Probability, Statistics
2
Replies
35
Views
500
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
591
Back
Top