Question

What is the De Morgan’s law for quantifiers?

(b) Write the negation of following statements.

(i) ∃x (x

2 + 2 < 1)

(ii) ∀x (x - 2 ≥ 3)

(iii) ∀x (x

2 ≥ 0 and x + 2 < 1)

(iv) ∃x (x

2 + 2 < 0 or x - 5 ≥ 0)

(v) ∀x (x - 2 ≥ 3 and x

3 + 5 ≤ 2)

(vi) ∀x (x

2 = 2)

(vii) There is a student in our class who likes chemistry.

(viii) Every fox is cunning.

Answer 1

a)

$\neg \forall x P(x)\iff\exist x\neg P(x)$

$\neg \exist x P(x)\iff\forall x\neg P(x)$

b)

i) $\forall x (x^2+2\geq1)$

ii) $\exist x (x-2<3)$

iii) $\exist x (x^2<0$ or $x+2\geq1)$

iv) $\forall x (x^2+2\geq0$ and $x-5<0)$

v) $\exist x (x-2<3$ or $x^3+5>2)$

vi) $\exist x (x^2\not=2)$

vii) All students in our class do not like chemistry.

viii) There exists a fox such that not cunning.