# 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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