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Dragonfall
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There is another PF member, poutsos.A, with the exact same prose as evagelos. They usually mutually agree on each other's posts. I wonder...
Proof of ΦxΦ = Φ and the special case of \emptyset\times\emptyset =\emptyset
- Context: Graduate
- Thread starter evagelos
- Start date
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SUMMARY
The discussion centers on the proof that the Cartesian product of the empty set with itself, denoted as \emptyset\times\emptyset, equals the empty set (\emptyset). Participants argue over the validity of various proof methods, including direct proofs and proofs by contradiction. A formal proof is presented, relying on established theorems in set theory, specifically that for any sets A and B, A\times B = \emptyset if and only if A=\emptyset or B=\emptyset. The conversation highlights the importance of clarity and rigor in mathematical proofs, particularly in the context of formal proofs versus informal reasoning.
PREREQUISITES- Understanding of set theory concepts, particularly the empty set.
- Familiarity with Cartesian products of sets.
- Knowledge of proof techniques, including direct proofs and proofs by contradiction.
- Basic understanding of logical operators and quantifiers in mathematical logic.
- Study the formal proof structure in set theory, focusing on axioms and theorems related to the empty set.
- Learn about the Mizar system for formal proofs in mathematics.
- Explore the Metamath project for detailed formal proofs and logical frameworks.
- Review the principles of proof by contradiction and its applications in set theory.
Mathematicians, students of mathematics, and educators interested in formal proof techniques and the foundational aspects of set theory.
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