Properties of real numbers

Assignment Help Engineering Mathematics
Reference no: EM13833103

1. Use the field axioms and their consequences to prove the following properties of real numbers. Your proofs should be careful and complete, with only one axiom, theorem, or other justification per step.

(a)  For any two real numbers a and b, we have that if a + b = 0, then b = -a. (Uniqueness of the additive inverse.)

(b)  For all a ∈ R, we have (-1)a = -a.

(c)  (-1). (-1) = 1.

(d)  For all a, b∈ R, we have (-a). (-b) = a b.

2. Use the field axioms, order axioms, and their consequences to prove the following properties of real numbers. (Algebraic properties will still need to be carefully proved/justified as you use them.)

(a)  For all a, b, c, d ∈ R, if a < b andc≤ d, then a + c< b + d.

(b)  If a, b ∈ R with 0 < a < b, then lib < 1/a.

3. Warm-up problems. These problems are just meant to be quick warm-ups to get you thinking about the definitions introduced in class and the reading. You do not need to justify anything or prove your answers.

(a) -(c) Find the infima and suprema of the set A = {1 + (-2/3)n ¦n ∈ N}, the function B given by B(x) = 4x - T2, and the set C = {1/n - 1/m ¦n, m ∈ N}.

(4) Give an example of a finite set D ⊆ R with sup D = -7 and inf D = -20.

4. State and prove a counterpart to Lemma 2.3.4 for inf S.

5. Suppose T ⊆ S ⊆ R are nonempty and bounded. Prove inf S ≤ muff' ≤ sup T ≤ sup S. Hint: Prove each inequality separately.

Reference no: EM13833103

Questions Cloud

The difference of one less than a number and twice number : the difference of one less than a number and twice the number
Evaluate the indefinite integral : Evaluate the indefinite integral.
Modern western worldview and the postmodern worldview : What is the main difference between the modern Western worldview and the postmodern worldview? Which parts, if any, do you identify with? Explain.
Identify where you might find hazard in industry : Identify where you might find this hazard either in industry or use in commercial products. Identify an occupation where someone might be overexposed to this hazard.
Properties of real numbers : 1. Use the field axioms and their consequences to prove the following properties of real numbers. Your proofs should be careful and complete, with only one axiom, theorem, or other justification per step.
What have you learned about being human : "What have you learned about being human in the last eleven weeks? More importantly, how will what you have learned impact your own life and the lives of those people around you?"
Set up to the equiation find the length of a side : set up to the equiation find the length of a side
Discuss economic strategy that you think help to close gap : Explain the sources of economic inequality (‘the gap') between indigenous and non-­-indigenous Australians and discuss an economic strategy that you think would help to close the gap.
The existence of dark matter or dark energy : Dark matter and dark energy are two of the biggest mysteries in science today - Identify one piece of evidence for the existence of either dark matter or dark energy.

Reviews

Write a Review

Engineering Mathematics Questions & Answers

  Prime number theorem

Dirichlet series

  Proof of bolzano-weierstrass to prove the intermediate value

Every convergent sequence contains either an increasing, or a decreasing subsequence.

  Antisymmetric relations

How many relations on A are both symmetric and antisymmetric?

  Distributed random variables

Daily Airlines fies from Amsterdam to London every day. The price of a ticket for this extremely popular flight route is $75. The aircraft has a passenger capacity of 150.

  Prepare a system of equations

How much money will Dave and Jane raise for charity

  Managing ashland multicomm services

This question is asking you to compare the likelihood of your getting 4 or more subscribers in a sample of 50 when the probability of a subscription has risen from 0.02 to 0.06.]  Talk about the comparison of probabilities in your explanation.

  Skew-symmetric matrices

Skew-symmetric matrices

  Type of taxes and rates in spokane wa

Describe the different type of taxes and their rates in Spokane WA.

  Stratified random sample

Suppose that in the four player game, the person who rolls the smallest number pays $5.00 to the person who rolls the largest number. Calculate each player's expected gain after one round.

  Find the probability density function

Find the probability density function.

  Develop a new linear programming for an aggregate production

Linear programming applied to Aggregate Production Planning of Flat Screen Monitor

  Discrete-time model for an economy

Discrete-time model for an economy

Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd