Question

In: Advanced Math

Using field axioms and order axioms prove the following theorems (explain every step by referencing basic...

Using field axioms and order axioms prove the following theorems (explain every step by referencing basic axioms)

(i) The sets R (real numbers), P (positive numbers) and [1, infinity) are all inductive

(ii) N (set of natural numbers) is inductive. In particular, 1 is a natural number

(iii) If n is a natural number, then n >= 1

(iv) (The induction principle). If M is a subset of N (set of natural numbers) then M = N

The following definitions are given:

A subset S of R is called inductive, if 1 is an element of S and if x + 1 is an element of S whenever x is an element of S.

The intersection of all inductive sets if called the set of natural numbers and is denoted by N

Solutions

Expert Solution

By the definition of Inductive given here, I have proved the four theorems. Hope you like the solutions. If you have any doubt then please leave a comment about that. I'll try my level best to solve your doubt.and according to my knowledge the fourth is true only if M is inductive. If you think this is wrong please share the picture of the page where this question is written. If you like the solution please give a thumbs up. Thanks in advance.


Related Solutions

Using field axioms, prove the following theorems: (i) If x and y are non-zero real numbers,...
Using field axioms, prove the following theorems: (i) If x and y are non-zero real numbers, then xy does not equal 0 (ii) Let x and y be real numbers. Prove the following statements 1. (-1)x = -x 2. (-x)y = -(xy)=x(-y) 3. (-x)(-y) = xy (iii) Let a and b be real numbers, and x and y be non-zero real numbers. Then a/x + b/y = (ay +bx)/(xy)
please prove this problem step by step. thanks Prove that in every simple graph there is...
please prove this problem step by step. thanks Prove that in every simple graph there is a path from every vertex of odd degree to some other vertex of odd degree.
Assume B is a Boolean Algebra. Prove the following statement using only the axioms for a...
Assume B is a Boolean Algebra. Prove the following statement using only the axioms for a Boolean Algebra properties of a Boolean Algebra. Uniqueness of 0: There is only one element of B that is an identity for + please include all the steps.
Prove using only the axioms of probability that if A and B are events, then P(A...
Prove using only the axioms of probability that if A and B are events, then P(A ∪ B) ≤ P(A) + P(B)
Prove the following theorem. Using the ruler function axiom. List all axioms and definitions used. Let...
Prove the following theorem. Using the ruler function axiom. List all axioms and definitions used. Let P and Q be two points, then the line segment AB=BA (AB and BA have lines over them to show line segments)
Using the axioms of probability, prove: a. P(A U B) = P(A) + P(B) − P(A...
Using the axioms of probability, prove: a. P(A U B) = P(A) + P(B) − P(A ∩ B). b. P(A) = ∑ P(A | Bi) P(Bi) for any partition B1, B2, …, Bn.
Prove: Every root field over F is the root field of some irreducible polynomial over F....
Prove: Every root field over F is the root field of some irreducible polynomial over F. (Hint: Use part 6 and Theorem 2.)
Prove both of the following theorems in the context of Incidence Geometry. Your proofs should be...
Prove both of the following theorems in the context of Incidence Geometry. Your proofs should be comparable in terms of rigor and precision (and clarity of thought!) to the ones done in class today. A1. Given any point, there is at least one line not passing through it, A2. Given any point, there are at least two lines that do pass through it,
Prove that every finite integral domain is a field. Give an example of an integral domain...
Prove that every finite integral domain is a field. Give an example of an integral domain which is not a field. Please show all steps of the proof. Thank you!!
1. What is a monotone class? 2.Prove that every algebra or field is a monotone class...
1. What is a monotone class? 2.Prove that every algebra or field is a monotone class Proof that 1. show that the intersection of any collection of algebra or field on sample space is a field 2. Union of field may not be a field
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT