Answer to Question #121259 in Discrete Mathematics for Kavee

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.