- #1
sutupidmath
- 1,630
- 4
Prove that a+b=b+a?
Hi all,
I am learning the axiomatic construction of Naturals, from the Peano axiom. Hence proving every property of naturals based solely on these axioms. I have come to some point where i am not very sure that what i am doing is okay, since there is not an explicit proof of this, it is left as an exercise to the reader.
I am trying to prove that for every a,b from naturals, a+b=b+a.
Here it is how i am going about it
[tex] S=\{b\in N | a+b=b+a\}[/tex] where a is any fixed natural numeber.
My attempt is to show that S is an inductive set.
We first need to prove that
[tex] 1\in S(?)[/tex]
Here it is how i go about it, this is the part that i am not sure,
Using another theorem, where it says that a+1=a', where a' is the successor of a...this in fact is the theorem where they define the operation '+'. But to prove this, the author defines this: 1+b=b' for every b in N.------(1)
Now on my proof above, i am not using the definition of '+' but rather this other definition 1+b=b' for every b in N. ... so my question is am i allowed to do this, or there is another way of going about it.
Here is how i go about it
a+1=a' from the definition 0f +
=1+a from definition (1)
this actually means that 1 is an element of S. the rest i can easily prove if this is correct.
So?
thnx
Hi all,
I am learning the axiomatic construction of Naturals, from the Peano axiom. Hence proving every property of naturals based solely on these axioms. I have come to some point where i am not very sure that what i am doing is okay, since there is not an explicit proof of this, it is left as an exercise to the reader.
I am trying to prove that for every a,b from naturals, a+b=b+a.
Here it is how i am going about it
[tex] S=\{b\in N | a+b=b+a\}[/tex] where a is any fixed natural numeber.
My attempt is to show that S is an inductive set.
We first need to prove that
[tex] 1\in S(?)[/tex]
Here it is how i go about it, this is the part that i am not sure,
Using another theorem, where it says that a+1=a', where a' is the successor of a...this in fact is the theorem where they define the operation '+'. But to prove this, the author defines this: 1+b=b' for every b in N.------(1)
Now on my proof above, i am not using the definition of '+' but rather this other definition 1+b=b' for every b in N. ... so my question is am i allowed to do this, or there is another way of going about it.
Here is how i go about it
a+1=a' from the definition 0f +
=1+a from definition (1)
this actually means that 1 is an element of S. the rest i can easily prove if this is correct.
So?
thnx