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 + <10
which in symbols is:
[ xεA & ( yεB====> + <10)]
]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:
( x=z).......................................PT..... .............................................1
THE above means that for all z there exists a unique x such that x=z
( 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=====> <10.....................it can be shown as an exercise in algebra........................................... 12
<10..........................................(11), (12),L............................................ .............................................13
y=0 ======> <10..................................from 10 to 13 by using the rule of conditional proof......................................14
In the same way we can prove :
y=1 =====> <10............................................... ..............................................15
y=2 ======> <10............................................... ...............................................16
y=0 v y=1 v y=2 ======> <10......................(14),(15),(16),L......... .................................................. ...............17
<10............................................... .(9),(17),L....................................... ...............................................18
yεB ====> <10.....................Again by conditional proof rule from lines 7 to 18...........................................19
( yεB ====> <10)..............................(19),PT......... ........................................20
xεA & ( yεB ====> <10)..............................(6),(20),L...... .........................................21
[xεA & ( yεB ====> <10)].......................................(21),PT.... .................................................. .22
So far we have proved the existence of an xεA Such that for all yεB THEN <10
Next step is to prove uniqueness of x.For that we must prove the following:
{[xεA & ( yεB ====> <10] & [zεA & ( yεB ====> <10]=====> x=z}
I leave that proof to you