x

Printable View

- Oct 22nd 2008, 11:32 AMprincessleia[SOLVED] Propositional functions and quantifiers ...
x

- Oct 22nd 2008, 12:09 PMPlato
$\displaystyle \left( {\exists x} \right)\left( {\forall z} \right)\left[ {F(A,x) \wedge M(S,x) \wedge \left[ {F(A,z) \wedge M(S,z) \Rightarrow \left( {x = z} \right)} \right]} \right]$

- Oct 22nd 2008, 05:03 PMpoutsos.B

To express in quantifier notation that Al and Sue**have**exactly one child we write:

$\displaystyle \exists !y$[ F( A,y) & M( S,y)] ,which means there is a unique child (y) ,lets say ,of whom Al and Sue are the father and mother respectively:

Now that formula is equivalent to the following formula:

$\displaystyle \exists y$[ F( A,y) & M( S,y)]& $\displaystyle \forall y\forall z${[F( A,y) & M( S,Y)] & [ F( A,z) & M( S,z)]------> y=z}:

The 2nd formula means that Sue and Al have a child and their child is the only one (expressed by the 2nd part of the formula:$\displaystyle \forall y\forall z${[F( A,y) & M( S,Y)] & [ F( A,z) & M( S,z)]------> y=z}:

NOW if my 2nd formula is equivalent to that given by Plato ,one should be able to prove that.

I think it is a good exercise for**you**