# Thread: [SOLVED] Propositional functions and quantifiers ...

1. ## [SOLVED] Propositional functions and quantifiers ...

x

2. $\displaystyle \left( {\exists x} \right)\left( {\forall z} \right)\left[ {F(A,x) \wedge M(S,x) \wedge \left[ {F(A,z) \wedge M(S,z) \Rightarrow \left( {x = z} \right)} \right]} \right]$

3. Originally Posted by princessleia
Let F(x, y) be the propositional function "x is the father of y", and let M(x, y) be the propositional function "x is the mother of y". Using only these two propostional functions and quantifiers, express the statement "Al and Sue have exactly one child" symbolically.

I don't even know where to start with this one! Any ideas?

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