Answer to Question #259421 in Discrete Mathematics for King

Question

For each of the given statements:

1 - Express each of the statements using quantifiers and propositional functions.

2 - Form the negation of the statement so that no negation is to the left of the quantifier.

3 - Express the negation in simple English. (Do not simply use the words “it is not

the case that...”).

(a) Some drivers do not obey the speed limit.

(b) All Swedish movies are serious.

(c) No one can keep a secret.

(d) No monkey can speak French.

(e) There is someone in the class who does not have a good attitude.

Answer 1

Part a

Let x denote drivers

Predicate P(x) denotes "x obey the speed limit"

The given statement in logical expression is $Ǝ x ¬P(x)$

Negation of the given statement in logical expression is

$¬Ǝ x ¬P(x) ≡ ∀ x ¬(¬P(x))$

All drivers obey the speed limit. 

Part b

Let x denote Swedish movies.

Predicate P(x) denotes "x is serious"

The given statement in logical expression is $∀ x (P(x))$

Negation of the given statement in logical expression is

$¬∀ x P(x) ≡ Ǝ x ¬P(x)$

There is a Swedish movie that is not serious 

Part c

Let x denote a person

Predicate P(x) denotes "x can keep a secret"

The given statement in logical expression is $∀ x (¬P(x))$

Negation of the given statement in logical expression is

$¬∀ x P(x) ≡ Ǝ x ¬(P¬(x))\\ ≡ Ǝ x P(x)\\$

There is someone who can keep a secret. 

Part d

No monkey can speak French.

We write this statement as.

$∀x(M(x) → ¬F(x))\\ ∀x(¬M(x)∨ ¬F(x))$

Its negation is $∃x(M(x) ∧ F(x))$ .

There is a monkey who can speak French.

Part e

Let x denote the class member

Predicate P(x) denotes "x does have a good attitude"

The given statement in logical expression is $Ǝ x ¬P(x)$

Negation of the given statement in logical expression is

$¬Ǝ x ¬P(x) ≡ ∀ x ¬(¬P(x))$

All drivers obey the speed limit. 

All the class members have a good attitude