Using the definition of the binary product

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SUMMARY

The discussion centers on the definition of the binary product, specifically addressing the statement: "If A x B = empty set, then A = empty set or B = empty set." Participants emphasize the importance of understanding the definition of the Cartesian product A x B, which is defined as the set of all ordered pairs (a, b) where a is in A and b is in B. The consensus is that if A x B is empty, at least one of the sets A or B must indeed be empty, as the presence of elements in both sets is necessary to produce any ordered pairs.

PREREQUISITES
  • Understanding of set theory concepts
  • Familiarity with Cartesian products
  • Knowledge of indirect proof techniques
  • Basic mathematical logic
NEXT STEPS
  • Study the formal definition of Cartesian products in set theory
  • Explore indirect proof methods in mathematical reasoning
  • Investigate counterexamples in set theory
  • Learn about the implications of empty sets in mathematical proofs
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Mathematicians, students of mathematics, and educators looking to deepen their understanding of set theory and proof techniques.

mbcsantin
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Prove or find counterexamples:
If A x B = empty set then A = empty set or B = empty set.
 
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Well, you titled this "using the definition of the binary product"?
So what is the definition of A x B?

Then consider using an indirect proof: If neither A nor B is empty...
 

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