The union of a subset and its complement

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SUMMARY

The union of a subset S and its complement cS with respect to a set X is equal to X. This conclusion is established by selecting any element from X and demonstrating that it belongs to the union of S and cS. The definitions of "complement" and "union" are crucial in this proof. Despite its straightforward nature, this fundamental concept is often overlooked in academic literature.

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  • Understanding of set theory concepts, specifically subsets and complements.
  • Familiarity with the definitions of union and intersection in set operations.
  • Basic knowledge of mathematical proofs and logical reasoning.
  • Experience with notation used in set theory, such as S, cS, and X.
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  • Study the properties of set operations, focusing on union and complement.
  • Explore advanced topics in set theory, such as De Morgan's laws.
  • Learn about applications of set theory in computer science, particularly in database management.
  • Investigate the role of set theory in mathematical proofs and logic.
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Mathematicians, computer scientists, educators, and students seeking a deeper understanding of set theory and its foundational principles.

chipotleaway
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If S is a subset of X, and cS is the complement of S with respect to X, is the union of S and cS equal to X? Seems like a no-brainer but just want to be sure because I've yet to find a book that comments on this.
 
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Yes. You can pick any element of X and prove that it is in the union of S and cS. Use the definition of "complement" and "union"
 
Haha appropriate username, thanks. How come this isn't mentioned in most books? Seems like all the 'obvious' stuff is.
 

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