Question

Theorem 16.11 Let A be a set. The set A is infinite if and only
if there is a proper subset B of A

for which there exists a 1–1 correspondence f : A -> B.

Complete the proof of Theorem 16.11 as follows: Begin by
assuming that A is infinite.

Let a1, a2,... be an infinite sequence of distinct elements of A.
(How do we know such a sequence

exists?) Prove that there is a 1–1 correspondence between the set A
and the set B = A\{a1}.

Answer #1

Write down a careful proof of the following.
Theorem. Let (a, b) be a possibly infinite open
interval and let u ∈ (a, b). Suppose that f : (a, b) −→ R is a
function and that lim x−→u f(x) = L ∈ R. Prove that for every
sequence an −→ u with an ∈ (a, b), we have t

. Write down a careful proof of the following. Theorem. Let (a,
b) be a possibly infinite open interval and let u ∈ (a, b). Suppose
that f : (a, b) −→ R is a function and that for every sequence an
−→ u with an ∈ (a, b), we have that lim f(an) = L ∈ R. Prove that
lim x−→u f(x) = L.

Proposition 16.4 Let S be a non–empty finite set.
(a) There is a unique n 2 N1 such that there is a 1–1
correspondence from {1, 2,...,n} to S.
We write |S| = n. Also, we write |;| = 0.
(b) If B is a set and f : B ! S is a 1–1 correspondence, then B is
finite and |B| = |S|.
(c) If T is a proper subset of S, then T is finite and |T| <...

Prove Corollary 4.22: A set of real numbers E is closed and
bounded if and only if every infinite subset of E has a point of
accumulation that belongs to E.
Use Theorem 4.21: [Bolzano-Weierstrass Property] A set of real
numbers is closed and bounded if and only if every sequence of
points chosen from the set has a subsequence that converges to a
point that belongs to E.
Must use Theorem 4.21 to prove Corollary 4.22 and there should...

1.- Prove the intermediate value theorem: let (X, τ) be a
connected topological space, f: X - → Y a continuous transformation
and x1, x2 ∈ X with a1 = f (x1), a2 = f (x2) ( a1 different a2).
Then for all c∈ (a1, a2) there is x∈ such that f (x) = c.
2.- Let f: X - → Y be a continuous and suprajective
transformation. Show that if X is connected, then Y too.

Let F be a ﬁeld (for instance R or C), and let P2(F) be the set
of polynomials of degree ≤ 2 with coeﬃcients in F, i.e.,
P2(F) = {a0 + a1x + a2x2 | a0,a1,a2 ∈ F}.
Prove that P2(F) is a vector space over F with sum ⊕ and scalar
multiplication deﬁned as follows:
(a0 + a1x + a2x^2)⊕(b0 + b1x + b2x^2) = (a0 + b0) + (a1 + b1)x +
(a2 + b2)x^2
λ (b0 +...

Let S denote the set of all possible finite binary strings, i.e.
strings of finite length made up of only 0s and 1s, and no other
characters. E.g., 010100100001 is a finite binary string but
100ff101 is not because it contains characters other than 0, 1.
a. Give an informal proof arguing why this set should be
countable. Even though the language of your proof can be informal,
it must clearly explain the reasons why you think the set should...

1. [10 marks] We begin with some mathematics regarding
uncountability. Let N = {0, 1, 2, 3, . . .} denote the set of
natural numbers.
(a) [5 marks] Prove that the set of binary numbers has the same
size as N by giving a bijection between the binary numbers and
N.
(b) [5 marks] Let B denote the set of all infinite sequences
over the English alphabet. Show that B is uncountable using a proof
by diagonalization.

Let A be a finite set and let f be a surjection from A to
itself. Show that f is an injection.
Use Theorem 1, 2 and corollary 1.
Theorem 1 : Let B be a finite set and let f be a function on B.
Then f has a right inverse. In other words, there is a function g:
A->B, where A=f[B], such that for each x in A, we have f(g(x)) =
x.
Theorem 2: A right inverse...

Consider an axiomatic system that consists of elements in a set
S and a set P of pairings of elements (a, b) that satisfy the
following axioms:
A1 If (a, b) is in P, then (b, a) is not in P.
A2 If (a, b) is in P and (b, c) is in P, then (a, c) is in
P.
Given two models of the system, answer the questions below.
M1: S= {1, 2, 3, 4}, P= {(1, 2), (2,...

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