Set theory: more basic problems to solve and clear and apply

I am producing the list of questions first so that the motivated reader can first read and try them …and can compare with my answers by scrolling down only much later; here we go:

  1. If \{ A_{i}\} and \{ B_{j}\} are two classes of sets such that \{ A_{i}\} \subseteq \{ B_{j}\}, then prove that \bigcup_{i}A_{i} \subseteq \{ B_{j}\} and \bigcap_{j} B_{j} \subseteq \bigcap_{i}A_{i}.
  2. The difference between two sets A and B, denoted by A-B, is the set of elements in A and not in B; thus, A - B = A \bigcap B^{'}. Prove the following simple properties: (a) A-B = A-(A \bigcap B) = (A \bigcup B)-B; (b) (A-B)-C = A-(B \bigcup C); (c) A - (B-C) = (A-B) \bigcup (A \bigcap C); (d) (A \bigcup B) - C = (A-C) \bigcup (B-C); (e) A - (B \bigcup C) = (A-B) \bigcap (A-C)
  3. The symmetric difference of two sets A and B, denoted by A \triangle B, is defined by A \triangle B = (A-B) \bigcup (B-A); it is thus the union of their differences in opposite orders. Prove the following : (a) Symmetric difference is associative : A \triangle (B \triangle C) = (A \triangle B) \triangle C (b) A \triangle \phi=A (c) A \triangle A = \phi (c) Symmetric difference is commutative: A \triangle B = B \triangle A (d) Some sort of distributive rule also holds: A \bigcap (B \triangle C) = (A \bigcap B) \triangle (A \bigcap C)
  4. A ring of sets is a non-empty class A of sets such that if A and B are in A, then A \triangle B and A \bigcap B are also in A. Show that A must also contain the empty set, A \bigcup B, and A-B. Show that if a non-empty class of sets contains the union and difference of sets any pair of its sets, then it is a ring of sets. Prove that a Boolean algebra of sets is a ring of sets.
  5. Show that the class of all finite subsets (including the empty set) of an infinite set is a ring of sets but is not a Boolean algebra of sets.
  6. Show that the class of all finite unions of closed-open intervals on the real line is a ring of sets but is not a Boolean algebra of sets.
  7. Assuming that the universal set U is non-empty, show that Boolean algebras of sets can be described as a ring of sets which contain U.

I will put up my solutions as soon as I can.

Regards,

Nalin Pithwa.

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