Question

Theorem 16.11 Let A be a set. The set A is infinite if and only if...

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}.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Write down a careful proof of the following. Theorem. Let (a, b) be a possibly infinite...
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...
. 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...
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...
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...
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...
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...
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,...
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....
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...
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,...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT