Proof of Venn Diagram for Sets A, B & C

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SUMMARY

The proof for the equation A complement U B complement = (A Intercept B) complement is established through a point-wise argument involving elements of both sides of the equation. By letting x be an element of the left-hand side (LHS) and demonstrating its implication on the right-hand side (RHS), and vice versa, the proof is validated. The discussion emphasizes the utility of Venn diagrams, which were developed by John Venn to visually represent the relationships between sets, enhancing the understanding of unions and intersections.

PREREQUISITES
  • Understanding of set theory concepts such as unions, intersections, and complements.
  • Familiarity with Venn diagrams and their applications in visualizing set relationships.
  • Basic knowledge of logical proofs and point-wise argumentation.
  • Awareness of historical context regarding John Venn and the evolution of diagrammatic representations of sets.
NEXT STEPS
  • Study the formal definitions of set operations, including union, intersection, and complement.
  • Learn how to construct and interpret Venn diagrams for multiple sets.
  • Explore logical proof techniques, particularly point-wise arguments in set theory.
  • Research the historical development of set theory and the contributions of John Venn.
USEFUL FOR

Mathematicians, educators, students of mathematics, and anyone interested in understanding set theory and its visual representations through Venn diagrams.

leilei
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Proof for all sets A, B, and C:
A complement U B complement = (A Intercept B) complement.

can someone help??
 
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Just to the typical point-wise argument. Let x be an element of the LHS and show that implies x is an element of the RHS. Then let x be an element of the RHS and show that implies x is an element of the LHS.

As you've already realized, you can draw a venn diagram (Who was this 'venn' by the way? I've always wondered that) to help yourself connect the dots.
 
rodigee said:
As you've already realized, you can draw a venn diagram (Who was this 'venn' by the way? I've always wondered that) to help yourself connect the dots.

John Venn was a logician who took the standard Euler Diagrams and improved upon them by overlapping and requiring that the overlap be the Union, and that all elements of the sets be constrained to exist in some area of the diagram.

Euler diagrams did not have to overlap, the circles could exist separate from each other. Venn's improvement assisted the logicians with understanding what the Union and disjunctions of sets meant.

Mathematicians then realized their usefulness began using them as well.

Wiki does both okay, but not great:
http://en.wikipedia.org/wiki/John_venn
http://en.wikipedia.org/wiki/Venn_diagram
 

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