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The Cut Property of the real numbers is the following: If A and B are nonempty, disjoint sets with A∪B = R and a < b for all a ∈ A and b ∈ B, then there exists c ∈ R such that x ≤ c whenever x ∈ A and x ≥ c whenever x ∈ B. (a) Use the Axiom of Completeness to prove the Cut Property. (b)Show that the implication goes the other way; that is, assume R possesses the Cut Property and let E be a nonempty set that is bounded above. Prove sup E exists. (c)The punchline of parts (a) and (b) is that the Cut Property could be used in place of the Axiom of Completeness as the fundamental axiom that distinguishes the real numbers from the rational numbers. To drive this point home, give a concrete example showing that the Cut Property is not a valid statement when R is replaced by Q.
asked
Dec 12, 2024
Mathematics
high-school
1
answer
3
votes
68.6k
views
Which point would be a solution to the system of linear inequalities shown below? 3 y -x = 2 y +6 a.(-4,-2) b. (-6,8) c.(-2,10) d. (12,-2)
asked
Dec 6, 2024
Mathematics
high-school
1
answer
3
votes
90.4k
views
Read the excerpt from act 3, scene 2 of The Tragedy of Macbeth. Lady Macbeth is speaking to herself. She has just sent her servant to tell Macbeth that she would like to speak to him. Lady Macbeth. Nought’s
asked
Nov 9, 2024
English
college
2
answers
5
votes
1.0k
views
Upon three or more convictions of operating a vehicle without proper supervision, a MA Junior Operator's permit will be suspended for a period of:
asked
Nov 8, 2024
Engineering
college
1
answer
2
votes
138k
views
-2 1/3 divided by 6 pls help
asked
Aug 28, 2024
Mathematics
high-school
1
answer
2
votes
224k
views
The word of the Clan leader is the warrior code. 1. True. 2. False.
asked
Jul 12, 2024
Social Studies
high-school
1
answer
5
votes
174k
views
A 41 year old male patient comes to the clinic with complaints of midback pain. Upon examination, the NP notices that the patient is guarding and has tenderness over the epigastric area. Cullen's sign
asked
Jun 22, 2024
Medicine
college
1
answer
0
votes
204k
views
Select all the stories that can be Represented by the equation, if none of the stories can be represented like none of the above
asked
Apr 2, 2024
Mathematics
high-school
1
answer
2
votes
133k
views
Which of the following is true of executive function? A) It shows considerable increase in late adulthood and is strongly linked to increase in working memory. B) It consists of a number of low-level cognitive
asked
Jan 23, 2024
Social Studies
high-school
2
answers
4
votes
181k
views
2 differences between system and applications software
asked
Dec 22, 2023
Computers and Technology
college
1
answer
6
votes
204k
views
Find the standard form of the equation for the circle with the following properties. Endpoints of a diameter are (-6,5) and (-2,-7)
asked
Oct 24, 2023
Mathematics
college
1
answer
3
votes
194k
views
Which expression is a trinomial
asked
Jan 9, 2023
Mathematics
high-school
2
answers
13
votes
234k
views
The sum of the squares of three consecutive integer numbers is 1454. Find the numbers. I have already found 21 22 23 but the question says that there are 2 possibilities. please help!!!!!!
asked
Oct 23, 2022
Mathematics
middle-school
1
answer
3
votes
134k
views
What motivated germany to sign a nonaggresion agreement with the soviet union
asked
Sep 21, 2022
History
high-school
1
answer
14
votes
120k
views
A bag contains 24 coins. Some of these are 10 cent coins and all of the others are 5 cent coins. iii) A second bag also contains 24 coins. in this bag the number of 5 cent coins ix x and all of the others
asked
Nov 22, 2021
Mathematics
high-school
1
answer
3
votes
136k
views
Why do you think the 1920's were called roaring?
asked
Sep 28, 2021
History
high-school
2
answers
3
votes
131k
views
Desmond and Molly each purchased 5 boxes of tangerines, and each box contains 60 tangerines. How many tangerines do they have in total?
asked
Aug 22, 2021
Mathematics
college
1
answer
2
votes
39.5k
views
The fossil evidence found in these sedimentary layers BEST implies that A) land plants evolved from algae. B) land plants evolved after marine animals. C) vertebrates and invertebrates evolved together.
asked
Jul 28, 2021
Social Studies
middle-school
2
answers
1
vote
4.1k
views
Which of the following best states the central disagreement between Abraham Lincoln and Stephen Douglas on the issue of slavery? whether it should be abolished whether it should be discouraged whether
asked
Jul 22, 2021
History
college
2
answers
3
votes
81.5k
views
How did the Supreme Court ruling in Dred Scott v Sandford influence the increasing cultural gap between the North and the South?
asked
Jun 27, 2021
History
high-school
1
answer
5
votes
168k
views
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