Proving Set Theory Union in Cartesian Products

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SUMMARY

The discussion centers on proving the set theory identity A × (B ∪ C) = (A × B) ∪ (A × C). Participants emphasize using the inclusion method to demonstrate the equality by analyzing the definitions of Cartesian products and unions. The key approach involves taking an element x from A × (B ∪ C) and showing that it must also belong to (A × B) ∪ (A × C) through logical reasoning based on set definitions.

PREREQUISITES
  • Understanding of set theory concepts, specifically Cartesian products and unions.
  • Familiarity with mathematical proofs, particularly the inclusion method.
  • Knowledge of basic set notation and operations.
  • Ability to manipulate and reason with elements of sets.
NEXT STEPS
  • Study the definitions and properties of Cartesian products in set theory.
  • Learn about the inclusion method for proving set identities.
  • Explore examples of set operations to solidify understanding of unions and intersections.
  • Practice proving other set theory identities using similar methods.
USEFUL FOR

Students studying set theory, mathematicians interested in foundational proofs, and educators teaching concepts of Cartesian products and unions.

rallycar18
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Homework Statement



Suppose A,B,C are sets. Prove that

A× (B U C)= (AxB) U (C x A)
 
Last edited:
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Have you tried the usual inclusion both ways method?
 
VeeEight said:
Have you tried the usual inclusion both ways method?

I'm not familiar..
 
Well if A is contained in B and B is contained in A, then A=B.
 
assume x \in A \times (B \cup C). Think about the definitions of Cartesian product and union: what can you conclude about the element x; can you use this information to show x \in (A \times B) \cup (A \times C)?
 

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