Can Disjoint Sets Help Prove Subset Relations in Set Theory?

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SUMMARY

The discussion focuses on proving that if sets A and B\C are disjoint, then A ∩ B ⊆ C. The user outlines their approach, starting with the premise that A ∩ B\C = ∅ and aims to demonstrate that A ∩ B must be a subset of C. Key points include the use of disjoint sets to derive contradictions and the necessity of understanding the implications of set membership. The conversation emphasizes logical reasoning and proof construction as essential skills in set theory.

PREREQUISITES
  • Understanding of set theory terminology, including disjoint sets and subset relations.
  • Familiarity with proof techniques, particularly proof by contradiction.
  • Knowledge of basic operations on sets, such as intersection and set difference.
  • Experience with logical reasoning as presented in "How to Prove It" by Daniel J. Velleman.
NEXT STEPS
  • Study the properties of disjoint sets in set theory.
  • Learn about proof by contradiction techniques in mathematical proofs.
  • Explore the implications of set operations, particularly intersection and set difference.
  • Review examples of subset proofs to enhance understanding of logical deductions.
USEFUL FOR

Students of mathematics, particularly those studying set theory, logic, and proof construction. This discussion is beneficial for anyone looking to strengthen their understanding of subset relations and disjoint sets.

eddiep1993
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I am currently reading vellemans how to prove it for the purpose of being able to construct a proof on my own. I would like to carry on this knowledge to also help me out with spivaks calculus So the problem is:

Prove that if a and b\c are disjoint, then a\bigcapb\subseteqc.
1.goal: a\bigcapb\c=∅ → a\bigcapb\subseteqc

2. Givens:a\bigcapb\c=∅ Goal:a\bigcapb\subseteqc

3.Givens:a\bigcapb\c=∅ x\ina x\inb Goal: x\inc
This is as far as I got. I haveE no idea where to go from here and I feel the solution is starting right at me. I think it might have something to do with the fact that a and b\c are disjoints. This might be in the wrong section but I don't no where else to put it
 
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hi eddiep1993! welcome to pf! :smile:
eddiep1993 said:
… I think it might have something to do with the fact that a and b\c are disjoints.

that's right :smile:

if x ε a, then x not ε in b/c …

carry on from there :wink:​
 
eddiep1993 said:
3.Givens:a\bigcapb\c=∅ x\ina x\inb Goal: x\inc
Maybe: a\bigcapb\c=∅ x\ina x\inb x\notinc Goal: contradiction
 

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