. . . . . . . 1.) The sentence

Some dogs eat vegetables.

can be written symbolically as

$ ( \exists x)( Q(x) \wedge P(x) ) $ .

A denial is

$ \sim ( \exists x)( Q(x) \wedge P(x) ) $

is equivalent to

$( \forall x) \sim ( Q(x) \wedge P(x) ) $ . . . . . (by Theorem 1.3 b.)

is equivalent to

$( \forall x) ( \sim Q(x) \vee \sim P(x) ) $ . . . . . (by Theorem 1.2 b.)

is equivalent to

$( \forall x) ( P(x) \Rightarrow \sim Q(x) ) $ . . . . . (by Theorem 1.2 d.)

is equivalent to

If an animal is a dog, then it does not eat vegetables.

is equivalent to

No dogs eat vegetables.


. . . . . . . 2.) The sentence

All dogs chase cars.

can be written symbolically as

$ ( \forall x)( Q(x) \Rightarrow R(x) ) $ .

A denial is

$ \sim ( \forall x)( Q(x) \Rightarrow R(x) ) $

is equivalent to

$( \exists x) \sim ( Q(x) \Rightarrow R(x)) $ . . . . . (by Theorem 1.3 a.)

is equivalent to

$( \exists x) \sim ( R(x) \vee \sim Q(x)) $ . . . . . (by Theorem 1.2 d.)

is equivalent to

$( \exists x) ( \sim R(x) \vee Q(x)) $ . . . . . (by Theorem 1.2 c.)

is equivalent to

There are animals, which are dogs and don't chase cars.

is equivalent to

Some dogs don't chase cars.


. . . . . . . 3.) The sentence

No dogs chase cars.

can be written symbolically as

$ ( \forall x)( Q(x) \Rightarrow \sim R(x) ) $ .

A denial is

$ \sim ( \forall x)( Q(x) \Rightarrow \sim R(x) ) $

is equivalent to

$( \exists x) \sim ( Q(x) \Rightarrow \sim R(x) ) $ . . . . . (by Theorem 1.3 a.)

is equivalent to

$( \exists x) \sim ( \sim R(x) \vee \sim Q(x) ) $ . . . . . (by Theorem 1.2 d.)

is equivalent to

$( \exists x) ( R(x) \wedge Q(x) ) $ . . . . . (by Theorem 1.2 c.)

is equivalent to

There are animals, which are dogs and chase cars.

is equivalent to

Some dogs chase cars.


. . . . . . . 4.) The sentence

There are some animals, which chase cars but do not eat vegetables.

can be written symbolically as

$ ( \exists x)( R(x) \wedge \sim P(x) ) $ .

A denial is

$ \sim ( \exists x)( R(x) \wedge \sim P(x) ) $

is equivalent to

$( \forall x) \sim ( R(x) \wedge \sim P(x) ) $ . . . . . (by Theorem 1.3 b.)

is equivalent to

$( \forall x) ( \sim R(x) \vee P(x) ) $ . . . . . (by Theorem 1.2 b.)

is equivalent to

$( \forall x) ( R(x) \Rightarrow P(x) ) $ . . . . . (by Theorem 1.2 d.)

is equivalent to

If an animal chases cars, then it eats vegetables.



RETURN to problem set.


Next: About this document ...
Duane Kouba 2002-06-06