Proof in predicate calculus

  1. How do we prove in predicate calculus using the laws of universal end existential quantifiers,propositional calculus,and those of algebra the following??

    There exists a unique x, xε{ 2,4,6} such that if yε{ 0,1,2} then x[tex]^{2}[/tex]y<10.
    or in quantifier form:


    [tex]\exists !x[/tex][ xεA & [tex]\forall y[/tex](yεB------> x[tex]^{2}[/tex]y<10)]

    where A={ 2,4,6} and B={ 0,1,2}
     
  2. jcsd
  3. poutsos.A

    I will try to give a proof of the above problem without mentioning the laws of logic, theorems or axioms and definitions used,you will have to do that.

    By a theorem in predicate calculus (with equality) we have.

    [tex]\forall z\exists !x[/tex](x=z)................................................................................................................1

    and for z=2 we have

    [tex]\exists x[/tex](x=2)......................................................................................................................2

    drop the existential quantifier and

    x=2.....................................................................................................................3

    but x=2 ====> x=2 v x=4 v x=6..........................................................................................................................4

    and from 3 and 4 we have : x=2 v x=4 v x=6..............................................................5

    but xεA <====> x=2 v x=4 v x=6..........................................................................................................................6

    and from 5 and 6 we get: xεA......................................................................................................................7

    now let yεB......................................................................................................................................................8

    But yεB <====> y=0 v y=1 v y=2...........................................................................................................................9

    and from 8 and 9 we get: y=0 v y=1 v y=2.........................................................................................................................10

    Now let y=0.........................................................................................................................11

    but y=0===> y^2=0====>.x[tex]^{2}[/tex]y=0<10.........................................................................................................................12

    and hence y=0 =====> x[tex]^{2}[/tex]y<10.........................................................................................................................13

    in a similar way we prove .

    y=1 ====>x[tex]^{2}[/tex]y<10....................................................................................14

    y=2 =====>x[tex]^{2}[/tex]y<10......................................................................................15

    hence: y=0 v y=1 v y=2=======>x[tex]^{2}[/tex]y<10..........................................................16

    and from 10 and 16 we get: x[tex]^{2}[/tex]y<10.....................................................................17

    hence : yεB======>x[tex]^{2}[/tex]y<10...........................................................................18

    And introducing universal quantification: [tex]\forall y[/tex]( yεB====>x[tex]^{2}[/tex]y<10)................................................................................19

    And thus: xεA & [tex]\forall y[/tex]( yεB====>x[tex]^{2}[/tex]y<10)..........................................20

    And introducing existential quantification we get; [tex]\exists x[/tex][ xεA & [tex]\forall y[/tex]( yεB====>x[tex]^{2}[/tex]y<10)]..............................................................................21

    NOW for the uniqueness part you have to prove that.


    [tex]\forall x\forall w[/tex]{[ xεA & [tex]\forall y[/tex](yεΒ=====>x[tex]^{2}[/tex]y<10)] & [ wεA & [tex]\forall y[/tex](yεΒ=====>w[tex]^{2}[/tex]y<10)] =====> x=w}
     
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