Prove that if A and B are disjoint denumerable sets, then A UB is denumerable.
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- Label each of the following statements as either true or false. 2. for all nonempty sets A and B.Label each of the following statements as either true or false. The least upper bound of a nonempty set S is unique.Label each of the following statements as either true or false. Every upper bound of a nonempty set is a least upper bound.
- Label each of the following statements as either true or false. If a nonempty set contains an upper bound, then a least upper bound must exist in .Prove that the cancellation law for multiplication holds in Z. That is, if xy=xz and x0, then y=z.16. Let and define on by if and only if . Determine whether is reflexive, symmetric, or transitive.
- Label each of the following statements as either true or false. The Well-Ordering Theorem implies that the set of even integers contains a least element.Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.[Type here] 7. Let be the set of all ordered pairs of integers and . Equality, addition, and multiplication are defined as follows: if and only if and in , Given that is a ring, determine whether is commutative and whether has a unity. Justify your decisions. [Type here]