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Math Help - proof in pridicate calculus

  1. #1
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    proof in pridicate calculus

    given A={2,4,6} and B={0,1,2} how do we prove in predicate calculus using quantifier and propositional logic laws and those of algebra the following:

    There exists a unique x, xεA,such that if yεB then x^2+ y^2<10

    which in symbols is:

    \exists x![ xεA & \forall y( yεB====> x^2+ y^2<10)]
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  2. #2
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    Quote Originally Posted by archidi View Post
    given A={2,4,6} and B={0,1,2} how do we prove in predicate calculus using quantifier and propositional logic laws and those of algebra the following:
    There exists a unique x, xεA,such that if yεB then x^2+ y^2<10
    which in symbols is: \exists x![ xεA & \forall y( yεB====> x^2+ y^2<10)]
    I doubt you can really do this within propositional logic.
    However, it is easy to show the statement is true for x=2\in A and it is false for x=4\in A or x=6\in A.
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  3. #3
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    Quote Originally Posted by Plato View Post
    I doubt you can really do this within propositional logic.
    However, it is easy to show the statement is true for x=2\in A and it is false for x=4\in A or x=6\in A.
    Thank you but my lecturer's instructions are for the proof within the predicate calculus which includes as you very well pointed out the propositional calculus
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  4. #4
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    Quote Originally Posted by archidi View Post
    Thank you but my lecturer's instructions are for the proof within the predicate calculus which includes as you very well pointed out the propositional calculus
    Well I hope that will favor us with your lecturer’s account of a proof.
    Even though I have taught symbolic logic many times, this sort of question makes me understand why a majority of mathematicians loath logicians.
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  5. #5
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    ya you right, logicians sometimes are unreal.Unfortunately i cannot supply you with a proof ,not until he gets our answers which are due in three weeks.
    He keeps on pumping with new questions every week
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  6. #6
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    ]In the following proof ...L...will indicate a law of propositional logic.
    PT a theorem in predicate calculus.
    AT a theorem in Algebra.
    A an axiom , D a definition.
    a No in brackets,for e,g (2) will denote the line of application of a law or PT.

    Hence:

    \forall z\exists! x( x=z).......................................PT..... .............................................1

    THE above means that for all z there exists a unique x such that x=z

    \exists x( x=2)................................(1),.PT....... .................................................. ..................................2

    x=2............................................... .(2).PT........................................... .................................................. .3


    x=2 v x=4 v x=6.........................................(3),L. .................................................. .......................................4

    xεA <===> x=2 v x=4 v x=6............................................D in set theory.............................5

    xεA............................................... ..................(4),(5),L....................... .................................................. .....................6


    yεB............................................... .assumption....................................... .................................................. .7


    yεB <====> y=0 v y=1 v y=2.......................................D....... .................................................8


    y=0 v y=1 v y=2............................................... ..(7),(8),L....................................... ...................................9


    y=0............................................... ...........assumption............................. ..................................10


    y=0 ^ x=2............................................... .......(3),(10),L................................. .................................................. ..........11


    y=0 ^ x=2=====> \ x^2 +y^2<10.....................it can be shown as an exercise in algebra........................................... 12


    \ x^2 +y^2<10..........................................(11), (12),L............................................ .............................................13


    y=0 ======> \ x^2 +y^2<10..................................from 10 to 13 by using the rule of conditional proof......................................14


    In the same way we can prove :


    y=1 =====> \ x^2 +y^2<10............................................... ..............................................15


    y=2 ======> \ x^2 +y^2<10............................................... ...............................................16


    y=0 v y=1 v y=2 ======> \ x^2 +y^2<10......................(14),(15),(16),L......... .................................................. ...............17


    \ x^2 +y^2<10............................................... .(9),(17),L....................................... ...............................................18


    yεB ====> \ x^2 +y^2<10.....................Again by conditional proof rule from lines 7 to 18...........................................19


    \forall y( yεB ====> \ x^2 +y^2<10)..............................(19),PT......... ........................................20


    xεA & \forall y( yεB ====> \ x^2 +y^2<10)..............................(6),(20),L...... .........................................21


    \exists x[xεA & \forall y( yεB ====> \ x^2 +y^2<10)].......................................(21),PT.... .................................................. .22

    So far we have proved the existence of an xεA Such that for all yεB THEN \ x^2 +y^2<10

    Next step is to prove uniqueness of x.For that we must prove the following:



    \forall x\forall z{[xεA & \forall y( yεB ====> \ x^2 +y^2<10] & [zεA & \forall y( yεB ====> \ z^2 +y^2<10]=====> x=z}

    I leave that proof to you
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