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In the rules for set complements theorem, we prove that the complement of a union is the intersection of the complements, and we proved that the complement of an intersection was the union of the complements. These are often called De Morgan's laws. The next problem has you prove De Morgan's laws are true regardless of the number of sets involved in the union or intersection.

Problem 80: (DeMorgan's Laws For Sets)

Suppose that for each $j$ in some nonempty set $J$ that $A_j$ is a set, and also suppose that $B$ is a set. Pick one of the statements below and prove that it is true. The other is very similar.

  • $\ds B\setminus \left(\bigcup_{j\in J}A_j\right) = \bigcap_{j\in J}\left(B\setminus A_j\right) $
  • $\ds B\setminus \left(\bigcap_{j\in J}A_j\right) = \bigcup_{j\in J}\left(B\setminus A_j\right) $

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