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:
- If and are two classes of sets such that , then prove that and .
- The difference between two sets A and B, denoted by , is the set of elements in A and not in B; thus, . Prove the following simple properties: (a) ; (b) ; (c) ; (d) ; (e)
- The symmetric difference of two sets A and B, denoted by , is defined by ; it is thus the union of their differences in opposite orders. Prove the following : (a) Symmetric difference is associative : (b) (c) (c) Symmetric difference is commutative: (d) Some sort of distributive rule also holds:
- A ring of sets is a non-empty class A of sets such that if A and B are in A, then and are also in A. Show that A must also contain the empty set, , and . 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.
- 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.
- 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.
- 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.