Answer
a) All drivers obey the speed limit.
b) There is a Swedish movie that is not serious.
c) Somebody can keep a secret.
d) Everybody in this class has a good attitude.
Work Step by Step
a) Let the domain be drivers and P(x) mean " x obeys the speed limit ". We can rewrite the given statement as :
$$\exists x \neg P(x)$$
The negation is then by De Morgan's Law for Quantifiers and double negation law:
$$\neg (\exists x \neg P(x)) \equiv \forall x \neg(\neg P(x)) \equiv \forall x P(x)$$
This means : All drivers obey the speed limits.
b) Let the domain be Swedish movies and Q(x) means " x is serious".We can rewrite the given statement as :
$$\forall x Q(x)$$
The negation is then by De Morgan's Law for Quantifiers:
$$\neg (\forall x Q(x)) \equiv \exists x \neg Q(x)$$
This means : There is a Swedish movie that is not serious.
c)Let the domain be people and R(x) be " x can keep a secret ". We can rewrite the given statement as :
$$\neg (\exists x R(x))$$
The negation is then by De Morgan's Law for Quantifiers and double negation law:
$$\neg (\neg (\exists x R(x))) \equiv (\exists x R(x))$$
This means : Somebody can keep a secret.
d)Let the domain be people in this class and S(x) means " x has a good attitude ".
We can rewrite the given statement as :
$$\exists x \neg S(x)$$
The negation is then by De Morgan's Law for Quantifiers and double negation law:
$$\neg (\exists x \neg S(x)) \equiv \forall x \neg(\neg S(x)) \equiv \forall x S(x)$$
This means : Everybody in this class has a good attitude.
