Give an example of two uncountable sets A and B such that A − B is a) finite. b) countably infinite. c) uncountable.

Disprove the statement that every positive integer is the sum of the cubes of eight nonnegative integers.

Use the Schröder-Bernstein theorem to show that (0, 1) and [0, 1] have the same cardinality.

Show that if A is an infinite set, then it contains a countably infinite subset.

Determine, with justification, whether each of the following sets is finite, countably infinite, or uncountable.(b) $\{x \in \mathrm{Q} \mid 1<x<2\}$

Let R₁ and R₂ be the “divides” and “is a multiple of” relations on the set of all positive integers, respectively. That is, R₁ = {(a, b) | a divides b} and R₂ = {(a, b) | a is a multiple of b}. Find a) R₁ ∪ R₂. b) R₁ ∩ R₂. c) R₁ − R₂. d) R₂ − R₁. e) R₁ ⊕ R₂.

On the day of a child’s birth, a parent deposits $30,000 in a trust fund that pays 5% interest, compounded continuously. Determine the balance in this account on the child’s 25th birthday.

Explain why the set A is countable if and only if

$$|A| ≤ |Z^+|.$$

Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. a) the integers greater than 10 b) the odd negative integers c) the integers with absolute value less than 1,000,000 d) the real numbers between 0 and 2 e) the set

$$A × Z^+$$

where A = {2, 3} f) the integers that are multiples of 10

Determine whether each of these sets is countable or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. a) all bit strings not containing the bit 0. b) all positive rational numbers that cannot be written with denominators less than 4. c) the real numbers not containing 0 in their decimal representation. d) the real numbers containing only a finite number of 1s in their decimal representation.

Give an example of an uncountable set.

Show that a finite group of guests arriving at Hilbert’s fully occupied Grand Hotel can be given rooms without evicting any current guest.

Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set. a) the negative integers b) the even integers. c) the integers less than 100. d) the real numbers between 0 and 1/2. e) the positive integers less than 1,000,000,000. f) the integers that are multiples of 7.

What is the difference between a constructive and nonconstructive existence proof? Give an example of each.