Question

Rewrite each of the following sentences using logical connectives. Assume that each symbol!, x₀, n, x, B represents some fixed object.
(a) If f has a relative minimum at x₀ and if f is differentiable at x₀, then f′(x₀)=0..,
(b) If n is prime, then n = 2 or n is odd.,
(c) R is symmetric and transitive whenever R is irreflexive.,
(d) B is square and not invertible whenever det B = 0.,
(e) f has a critical point at x₀ iff f '(x₀) = 0 or f'(x₀) does not exist.,
(f) 2 < n - 6 is a necessary condition for 2n < 4 or n > 4.,
(g) 6 ≥ n - 3 only if n > 4 or n > 10.,
(h) x is Cauchy implies x is convergent.,
( i) f is continuous at x₀ whenever $lim_{x \rightarrow x_0}$ f(x) = f(x₀).,
(j) If f is differentiable at x₀ and f is increasing at x₀, then f'(x₀) > 0.

Answer 1

Answer:

a) $(f\ has\ a\ relative \ minimum \ at\ x_{0} )$ ∧ $(f\ is\ differenciable\ at\ x_{0} )$ ⇒ $f^{'} (x_{0} )=0$.

b) $(n\ is\ prime)$ ⇒ $(n=2)$ ∨ $(n\ is \ odd)$.

c) $R\ is \ irreflexive$ ⇒ $(R\ is\ symmetric)$ ∧ $(R\ is\ transitive)$.

d) $detB=0$ ⇒ $(B \ is\ square)$ ∧ $(B\ is\ not\ invertible)$.

e) $f\ has\ a\ critical\ point\ at \ x_{0}$ ⇔ $(f^{'}(x_{0})=0)$ ∨ $() \ f^{'}(x_{0}) \ does\ not\ exist$.

f) $2n$ ∨ $n>4$ ⇒ $2$.

g) $6\geq n-3$ ⇔ $n>4$ ∨ $n>10$.

h) $x \ is \ cauchy$ ⇒ $x\ is\ convergent$.

i) $\lim_{x \to \ x_{0}} f(x)=f(x_{0})$ ⇒ $f\ is\ continous\ at \ x_{0}$.

j) $(f\ is\ diferenciable\ at\ x_{0})$ ∧ $(f\ is\ increasing\ at\ x_{0})$ ⇒ $f^{'}(x_{0})>0$.