1. ## 1+1

In university, if the question ask:
Prove that 1+1=2.

How to prove?

2. Originally Posted by SengNee
In university, if the question ask: Prove that 1+1=2.
How to prove?
It depends upon your axiom set.

3. Originally Posted by SengNee
In university, if the question ask:
Prove that 1+1=2.

How to prove?
You need to first define "integer". That is what Plato said.

4. Having seen this before, if you were not given a set of axioms, then Peano's axioms may be helpful for this. They are:

1. Let S be a set such that for each element x of S there exists a
unique element x' of S.

2. There is an element in S, we shall call it 1, such that for every
element x of S, 1 is not equal to x'.

3. If x and y are elements of S such that x' = y', then x = y.

4. If M is any subset of S such that 1 is an element of M, and for
every element x of M, the element x' is also an element of M, then
M = S.

5. Originally Posted by galactus
Having seen this before, if you were not given a set of axioms, then Peano's axioms may be helpful for this. They are:

1. Let S be a set such that for each element x of S there exists a
unique element x' of S.
what are the properties of this element x'? how does it relate to x?

6. The way I have seen it done is thus:

As a matter of notation, we write 1' = 2, 2' = 3, etc. We define

$\displaystyle 1. \;\ x + 1 = x'$

$\displaystyle 2. \;\ x + y' = (x + y)'$

The element x + y is called the sum of x and y.

Now to prove that 1 + 1 = 2.

From 1: with x = 1, we see that 1 + 1 = 1' = 2.

Standard properties of addition - for example, x + y = y + x for all x
and y in S - can be proved by induction (which is based on Peano's
Postulate #4.

7. Originally Posted by SengNee
In university, if the question ask:
Prove that 1+1=2.

How to prove?
Alfred North Whitehead and Bertrand Russell wrote Principia Mathematica and published it in three volumes in the years 1910-1913. In it they laid the foundation of modern mathematics. On page 362, they finally got around to proving that 1 + 1 = 2.

On the other hand, 1 + 1 = 3 for 1 sufficiently large

8. Originally Posted by mr fantastic
Alfred North Whitehead and Bertrand Russell wrote Principia Mathematica and published it in three volumes in the years 1910-1913. In it they laid the foundation of modern mathematics. On page 362, they finally got around to proving that 1 + 1 = 2.
I remember reading that too. Most people do not realize how complicated simple addition is.

9. I still confuse. Anywhere, thanks.

10. Originally Posted by SengNee
I still confuse. Anywhere, thanks.
what axioms are you using? you have to tell us what you have at your disposal for us to properly help you

11. I only a student after SPM(only available in Malaysia, its standard is same as "O" level), waiting for result.
I post this because I hope to learn somethings extra.

12. Originally Posted by SengNee
I still confuse. Anywhere, thanks.
In the Peano system 1+1 is another name for the successor of 1 which
by convention is named 2, so 1+1=2 is saying nomore than 1+1=1+1.

RonL

13. Originally Posted by ThePerfectHacker
I remember reading that too. Most people do not realize how complicated simple addition is.
Most people don't, but they are assuming the basic axioms of mathematics.

14. Originally Posted by SengNee
I only a student after SPM(only available in Malaysia, its standard is same as "O" level), waiting for result.
I post this because I hope to learn somethings extra.
well, so far i think galactus' approach is the most rigorous, so you may want to note that. however, do give careful considerations to any other answer that was given

15. Thank you everyone.