Bifalgorithm post · Brisk · Chapter 1

Why do a set and its complement form a partition?

Why do a set and its complement form a partition? (Brisk · Chapter 1) — "Every set (S) together with its (total) complement (S) forms a partition of the entire…

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  • A set and its total complement form a partition of the domain of discourse Ω.
  • For any set S, the sets S and its complement together partition Ω.
  • Every element of Ω is either in S or in its complement.
  • No element can be in both S and its complement.
  • Males and non-males partition Ω, including women, inanimate objects, and other nouns among the non-males.
  • Prime numbers and everything except prime numbers partition Ω.
  • Underdone cheeseburgers and everything except underdone cheeseburgers partition Ω.
  • By pure logic, this partition property holds no matter what the set is.
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Why do a set and its complement form a partition?

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Brisk

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"Every set (S) together with its (total) complement (S) forms a partition of the entire domain of discourse Ω. This is because every element either is, or is not, in any given set. The set of males and non-males are a partition of Ω because everything is either a male or a non-male, and never both (inanimate objects and other nouns are non-males, just as women are). The set of prime numbers and the set of everything-except-prime-numbers are a partition. The set of underdone cheeseburgers and the set of everything-except-underdone-cheeseburgers form a partition of Ω. By pure logic, this is true no matter what the set is."

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