# Proving Properties of Natural NumbersWatch

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#1
Hi Guys

Basically, I'm working through the Analysis I Textbook by Terence Tao. For those who aren't familiar with it, basically it begins by starting with the Peano axioms and using these to deduce other properties of the natural numbers. I believe the idea is to get used to handling proofs by first proving things which seem obvious so that we can handle proving results in analysis which won't seem so obvious.

So, anyway, enough of my rambling. I was wondering if anyone could assist me in proving these statements and also advise whether I'm structuring my proofs in the correct way.

Below are the axioms, propositions, lemmas, etc. which we may assume:
Spoiler:
Show

Below are the exercises:
Spoiler:
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I'll include my working on each exercise in a different post to avoid crowding this one out

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#2
Exercise 2.2.1

Let be the proposition that for all natural numbers

Therefore P(0) is true.

Which is P(a++)

Therefore by the principle of mathematical induction (axiom 2.5), P(a) is true for all natural numbers, a.

---

Is this a valid structure for a proof?
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#3
I'm not sure how to approach exercise 2.2.2

I know it says to use induction but I'm not sure on what, a or b?
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#4
Exercise 2.2.3. a)

iff , for some natural number n.

by lemma 2.2.2 and 0 is a natural number according to axiom 2.1.

Therefore . QED

Is this a valid structure?
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#5
Exercise 2.2.3 b

iff , for some natural number (definition 2.2.11)

and iff , for some natural number (definition 2.2.11)

Therefore,

So has the form where is some natural number

Therefore it follows that from definition 2.2.11

QED.

Is this ok?
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#6
Exercise 2.2.3 c

iff for some natural number n

iff for some natural number m

Therefore, and iff
iff because of the cancellation law (proposition 2.2.6)
iff m = 0 and n = 0 by corollary 2.2.9

Therefore by lemma 2.2.2 and by lemma 2.2.2

QED
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#7
Exercise 2.2.3 d

iff for some natural number
iff

because addition is commutative and associative (propositions 2.2.4 and 2.2.5)

Therefore, it follows that, by definition 2.2.11

QED
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#8
Exercise 2.2.3 e

iff for some natural number n and

implies because lemma 2.2.2

So we can write, for some natural number m (I'm not sure if I've justified this enough?)

iff by definition 2.2.11

QED
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#9
Exercise 2.2.4

This is the proof provided by the book:
Spoiler:
Show

iff for some natural number n (from definition 2.2.11)
by definition 2.2.1
and we know that b is a natural number therefore for all natural numbers QED
(is this ok, or would I need to use induction?)

iff and

iff
(I'm not sure if this step is actually justified? Since axiom 2.4 just says n++ = m++ => n=m, not the other way around)
by lemma 2.2.3
axiom says if n is a natural number then so is n++ so
for some natural number m
iff

Axiom 2.3 says 0 is not the successor of any natural number and since we know
this means
iff
iff by lemma 2.2.2

So if and then QED

iff
iff by lemma 2.2.2
iff by definition 2.2.1
iff by lemma 2.2.3
iff (by definition 2.1.3)
iff by definition 2.2.11

suppose that
iff by the cancellation law (proposition 2.2.6)
but and axiom 2.3 says that 0 is not the successor of any natural numbers
therefore
so
so

We have and
therefore definition 2.2.11

QED
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#10

Anyone willing to look at these proofs and comment?
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