Show in detail that f is a bijection (Corollary's proof) ove that every infinite set S contains a proper subset similar to SClear searchTheorem 1.42. Every infinite set contains a countably infinite subset.Proof. Let A be an infinite set. Take some element of A and callit az. The set A-{az} is not empty; denote one of its elements by az.This process may be continued indefinitely: at the nth stage the setA-{a1, ..., an-1}cannot be empty since A is infinite. The set {a,, a„ ..} is a countablyinfinite subset of A. ||Corollary. Every infinite set contains a proper subset to which it issimilar.Proof. Let A be an infinite set and let S = {a, az, ..} be a count-ably infinite subset of A. The function f: A → A-{az} defined by%3D(an+1 if x = a, (n = 1, 2, ..),f(x)if x e A-Sis a bijection. Thus A ~ A-{az}. |

Basically, you are asking to show that the function f:A→A-{a1} defined as f(x)=an+1 if x=an…

Corollary. Every infinite set contains a proper subset to which it is similar. Proof. Let A be an infinite set and let S = {a, az, ..} be a count- ably infinite subset of A. The function f: A → A-{a;} defined by (an+1 if x = a, (n = 1, 2, ...), if x € A-S %3D f(x) = (x is a bijection. Thus A ~ A-{a,}. |

Corollary. Every infinite set contains a proper subset to which it is similar. Proof. Let A be an infinite set and let S = {a, az, ..} be a count- ably infinite subset of A. The function f: A → A-{a;} defined by (an+1 if x = a, (n = 1, 2, ...), if x € A-S %3D f(x) = (x is a bijection. Thus A ~ A-{a,}. |

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter2: Working With Real Numbers

Section2.1: Basic Assumptions

Problem 40WE

See similar textbooks

Related questions

Q: 4. If {X(t)} is a WSS process and if Y(t) = | h(u)X(t – u)du, then Ry,(T) = R„(T)* K(t), K(t) =…

Q: Show that if both f: A ---> B and g: B ----> C are bijective, then g 0 f: A ---> C is bijective.

Q: 2. If the member functions of a Gaussian process are uncorrelated, then they are independent.

Q: Supposer FCE, acE is algebraic F and dog (a/F) 13 odd. Proveo that F(a)= F(a²). over

Q: (3) P(A| B) > 0, i.e., the conditional probability is always non-negative.

Q: Hypothesis Testing for Two Proportions Preform a hypothesis test on a separate sheet of paper Nathan…

A: According to our guidelines, we have to sove only first problem.

Q: It states that if a sample space S has N mutually exclusive events B, n=1,2,....,N, such that, B,…

Q: If the characteristic function of a RV X is f,(@) and if Y = aX + b, then O,(@) = ibo e

Q: 2 94

A: according to question function is f(x)=x+4 x=1,2,3,4 f(1)=1+4=5 f(2)=2+4=6 f(3)=3+4=7

Q: evalnante the folobwing integrationollowing a fferent Path C I'm troid.cpathz pathi Re (3,-2) path3

A: Note:- For part (3) it is not clear that the term written with sint (or coefficient of sint ) is "I"…

Q: 5. If the number of occurrences of an event E is an interval of length t is a Poisson process {X(t)}…

Q: The spectral density function of a real random process is an even function.

Q: Derive at least one valid immediate conclusion in the statement and explain it's not the case that…

A: The given statement is:It's not the case that no F is G.The aim of the problem is to derive at least…

Q: 3. If (X(t)} is a WSS process and if 00 Y(t) = | h(u) X(t – u)du, then - 00 (i) Ry (t) = R(T)* h(-t)…

Q: How can I prove that f is one-to-one (injective)?

A: We know, a function/map f is one to one (injective) iff For a ≠ b, f(a) ≠ f(b) for any two…

Q: (5) If two events A and B are mutually exclusive, then the joint events, AnC and BOC are also…

Q: Use quotient rule to show that /dx(cscx)--cscx coty Show AU Wonc

Q: 1. If a Gaussian process is wide-sense stationary, it is also strict-sense stationary.

Q: The interarrival time of a Poisson process, i.e., the interval between two successive occurrences of…

Q: Additive property: Sum of two independent Poisson processes is a Poisson process.

Q: 4. If R(T) is the autocorrelation function of a stationary process {X(t)} with no periodic…

Q: Assume that ∀x∃yP(x, y) is false and the domain of them is nonempty. Which of the following must…

A: See the detailed solution below.

Q: The coefficient of correlation is independent of the change of origin and scale of measurements.

Q: We learned how to construct the map ² (V) → VV that sends the simple wedge V₁ V₂ → V₁ V₂ V₂ V₁ -…

A: Given that the map Λ2V→V⊗V is defined by the function, v1 Λ v2→v1⊗v2-v2⊗v1 We are asked to prove…

Q: Consider a sample space S that has N mutually exclusive events B, n=1,2,...,N , such that BB, = {Ø}…

Q: 1. The Poisson process is a Markov process.

Q: Let A = {0, 1, 2, ..., 7}. Define f : A→ A and g : A→ A by f(n) g(n) (5n + 2) mod8 (4n + 3) mod 8…

A: Given that A=0, 1, 2, ... , 7. Functions f:A→A and g:A→A are defined as fn=5n+2…

Q: Multiplication theorem can be used to find out the probability of outcomes when an experiment is…

Q: (b) If S is a set of content zero in RP, then its closure 5 (union of the boundary of S and interior…

Question

Show in detail that f is a bijection (Corollary's proof)

ove that every infinite set S contains a proper subset similar to S
Clear search
Theorem 1.42. Every infinite set contains a countably infinite subset.
Proof. Let A be an infinite set. Take some element of A and call
it az. The set A-{az} is not empty; denote one of its elements by az.
This process may be continued indefinitely: at the nth stage the set
A-{a1, ..., an-1}
cannot be empty since A is infinite. The set {a,, a„ ..} is a countably
infinite subset of A. ||
Corollary. Every infinite set contains a proper subset to which it is
similar.
Proof. Let A be an infinite set and let S = {a, az, ..} be a count-
ably infinite subset of A. The function f: A → A-{az} defined by
%3D
(an+1 if x = a, (n = 1, 2, ..),
f(x)
if x e A-S
is a bijection. Thus A ~ A-{az}. |

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 1 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

Determine whether the set S={2x+x2,8+x3,x2+x3,4+x2} spans P3.
Suppose a set A has 2,048 subsets. How many distinct objects are contained in A?
True or False Label each of the following statement as either true or false. 1. Parity check schemes will always detect the position of an error.
Express the set of real numbers between but not including 2 and 7 as follows. (a) In set-builder notation:__________________ (b) In interval notation:__________________
Assume that is a binary operation on a non empty set A, and suppose that is both commutative and associative. Use the definitions of the commutative and associative properties to show that [ (ab)c ]d=(dc)(ab) for all a,b,c and d in A.