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Math Help - 1+1

  1. #1
    Member SengNee's Avatar
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    1+1

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

    How to prove?
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  2. #2
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    Quote Originally Posted by SengNee View Post
    In university, if the question ask: Prove that 1+1=2.
    How to prove?
    It depends upon your axiom set.
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    Quote Originally Posted by SengNee View Post
    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.
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    Eater of Worlds
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    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.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by galactus View Post
    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?
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    Eater of Worlds
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    The way I have seen it done is thus:

    As a matter of notation, we write 1' = 2, 2' = 3, etc. We define
    addition in S as follows:

    1. \;\  x + 1 = x'

    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.
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  7. #7
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    Quote Originally Posted by SengNee View Post
    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
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    Quote Originally Posted by mr fantastic View Post
    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.
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  9. #9
    Member SengNee's Avatar
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    I still confuse. Anywhere, thanks.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by SengNee View Post
    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
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  11. #11
    Member SengNee's Avatar
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    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.
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    Quote Originally Posted by SengNee View Post
    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
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    GAMMA Mathematics
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    Quote Originally Posted by ThePerfectHacker View Post
    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.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by SengNee View Post
    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
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  15. #15
    Member SengNee's Avatar
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    Thank you everyone.
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