Proving Set Relationships: A-B ⊂ A & (A∩B)c = A∪Bc

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Discussion Overview

The discussion revolves around proving set relationships, specifically the inclusion of the set difference A-B within A and the complement relationships involving intersections and unions of sets A and B. The scope includes theoretical aspects of set theory and definitions related to subsets and complements.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants suggest starting with the definitions of set operations to approach the proofs, specifically mentioning the definition of the set difference A\B.
  • One participant emphasizes the need to understand what it means for A-B to be a subset of A, indicating that all elements of A-B must be contained in A.
  • Another participant expresses confusion about proving the statements without explicit definitions, highlighting the challenge of working with abstract concepts.
  • There is a reiteration of the idea that each element of A-B is indeed an element of A, which is a key point in understanding the proof.

Areas of Agreement / Disagreement

Participants generally agree on the need for definitions to understand and prove the set relationships, but there is some contention regarding the necessity of those definitions in the context of the problem presented.

Contextual Notes

Participants note the absence of explicit definitions in the problem statement, which may limit their ability to formulate a proof effectively. The discussion reflects a reliance on understanding the foundational concepts of set theory.

Chis96
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Can some please explain to me how to solve these two questions?
Let A, B and C be any sets, prove that:
(a) A-B ⊂ A
(b) (A∩B) complement = A complement ∪ B and (A∪B) complement = A complement∩ B complement.
 
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We usually write out A\B instead of A -B

Anyway, start to write down the definitions and see where you get.

A\B = {x|x ∈ A and x ∉ B}
 
Thank you very much for the correction.
Well... there are no definitions, it just says let A, B and C be any set, prove that A\B⊂A.
 
Well, look up the definitions! How can you prove something without knowing what the definitions are?
 
No definitions bro, just have to use x to prove it... That's why I'm confused.
 
Chis96 said:
No definitions bro, just have to use x to prove it... That's why I'm confused.

He's talking about the definition of "subset"! As in, what does ##A-B \subset A## actually mean? You can't prove it unless you know what it means.
 
oh... okay basically what it means is that all elements of A\B are contained inside A.
 
Chis96 said:
oh... okay basically what it means is that all elements of A\B are contained inside A.

Yes, although more simply and consistently you could say it means:

Each element of A\B is an element of A.
 
Yes, thank you very much sir.
 

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