Proving Properties of Natural Numbers
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LogicGoat
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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:
Below are the exercises:
I'll include my working on each exercise in a different post to avoid crowding this one out
Thanks for your time
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:
Show

I'll include my working on each exercise in a different post to avoid crowding this one out
Thanks for your time

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LogicGoat
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#2
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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?
I know it says to use induction but I'm not sure on what, a or b?

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LogicGoat
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#9
Exercise 2.2.4
This is the proof provided by the book:
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
This is the proof provided by the book:
Spoiler:
Show




and we know that b is a natural number therefore


(is this ok, or would I need to use induction?)



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)

axiom says if n is a natural number then so is n++ so

iff

Axiom 2.3 says 0 is not the successor of any natural number and since we know

this means

iff

iff

So if





iff

iff

iff

iff

iff

suppose that

iff

but

therefore

so

so

We have


therefore

QED
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#10
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