130 Chapter 2 • Sequences
- Prove that the relation ~ of Definition 2.8. l has the following properties:
(a) (Reflexivity) \IA, A~ A.
(b) (Symmetry) \IA, B, A~ B::::;. B ~A.
(c) (Transitivity) \IA, B, C, if A~ B and B ~ C, then A~ C.- Prove that if A is any infinite set and x E A, then A ~ A - { x}. [Hint:
Apply Theorem 2.8.4; make x the first element of a denumerable subset of
A and consider the function f(xk) = Xk+l on this subset, while f(x) = x
otherwise. J - Prove that if A is an infinite set and B is any finite subset of A, then
A~A-B. - Prove that if A is an infinite set, then there is some denumerable subset
B of A, such that A ~ A - B. - Prove that if A is an uncountable set and B is any countable subset of
A,thenA~A-B. - Suppose a< b, and c < d. Prove that (a, b) ~ (c, d) and [a , b] ~ [c, d] by
constructing 1-1 correspondences b etween the intervals.
11. Suppose a< b. Prove that (a, b) ~(a, +oo) by constructing a 1-1 corre-
spondence between the intervals.- Suppose a< b, and c < d. Prove that (a, b) ~ [c, d].
- Prove that (0, 1) ~ R [An interval is equivalent to the whole line!]
- Prove that (0 , 1) x (0, 1) ~ (0, 1). [Hint: Use decimal expansions.]
- Prove that if A~ C and B ~ D, then Ax B ~ C x D.
16. Prove that JR x JR ~ R [The plane is equivalent to a line!]- (Project) Algebraic and Transcendental Numbers: By definition,
an algebraic number is any real number that is a solution of a polyno-
mial equation p(x) = 0, where p(x) has integer coefficients. A transcen-
dental number is a real number that is not algebraic.
Assume that every algebraic number x satisfies a unique polynomial equa-
tion with rational coefficients of the form xn+an_ 1 xn-^1 + ... +a 1 x+a 0 = 0
of lowest degree. [You could prove this using the factor theorem and the
unique factorization theorem of algebra.] The degree of this polynomial