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Math Help - formalization of sqroot definition

  1. #1
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    formalization of sqroot definition

    Recently i was asked to formalize the following sqroot definition:

    We define the sqroot of a non negative No x ( x>=0),symbolized as sqroot(x), another non negative No y such that when we raise y to the power of two we get x.
    Real Nos of course
    Although i have an idea your ideas will be appreciated
    And then to prove the relevant theorems in a formalized manner
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  2. #2
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    My idea of the formula is the following:

    ..............(x)(y).[sqroot(x)=y <-----> y^2=x ]

    do you agree??
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  3. #3
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    Quote Originally Posted by triclino View Post
    Recently i was asked to formalize the following sqroot definition:

    We define the sqroot of a non negative No x ( x>=0),symbolized as sqroot(x), another non negative No y such that when we raise y to the power of two we get x.
    Real Nos of course
    Although i have an idea your ideas will be appreciated
    And then to prove the relevant theorems in a formalized manner
    To show this is a good definition you need to show such a number y is unique and it exists.
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  4. #4
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    Quote Originally Posted by ThePerfectHacker View Post
    To show this is a good definition you need to show such a number y is unique and it exists.
    Thank you,

    but do we have to do that for every definition??

    For example in the definition of the composite function do we have to show that it exists??

    In our case the existence has been shown by the following theorem:

    .......(x)[ x>=0---------> E!y( y>=0 & y^2=x)].........................

    Hence the definition of sqroot(x)

    Where E!y means there exists a unique y,and (x) for all x
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  5. #5
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    And now how do we prove :

    1) x>=0=======> sqroot(x) >=0

    2) x>=0 ======> [sqroot(x)]^2=x

    3) sqroot(x^2)= absvalue(x)

    4) x>=0 & y>=0=====> sqroot(xy) = sqroot(x)(y)

    by using only:............ sqroot(x) = y <------> y^2=x ????....................

    thanks
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