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

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Advance Math

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, 1is 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 ofnatural numbers) then M = N

The following definitions are given:

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

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

Answer & Explanation Solved by verified expert
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By the definition of Inductive given here I have proved thefour theorems Hope you    See Answer
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