2.3 a)
-
(a) $$ (\exists x \in \mathbb{N}) (x^3=27)$$
(b) $$ (\exists p \in \mathbb{N}) (p > 1,000,000) $$
(c) $$ \exists((p \in \mathbb{N})\wedge (1<p<n)) \frac{n}{p}\in \mathbb{N} $$ -
(a) $$ (\forall x \in \mathbb{N})(x^3 \neq 8) $$
(b) $$ (\forall p \in \mathbb{N})(0 < p) $$
(c) $$ \forall(( p \in \mathbb{N})\wedge (1<p<n)) \frac{n}{p}\in \mathbb{N} $$ -
(a) For all the people, it is the case that there exists someone they love:
$$ (\forall x \in \text{people}) (\exists y \in \text{people}) (x \text{ loves } y) $$
(b) $$ (\forall x \in \text{people})(x \text{ is tall}) \text{ or } (\forall x \in \text{people}) (x \text{ is short}) $$
(c) $$ (\forall x \in \text{people}) ((x \text{ is tall}) \text{ or } (x \text{ is short})) $$
(d) $$ (\forall x \in \text{people}) (x \text{ is not at home}) $$
(e) $$ (\forall x \in \text{men}) (\forall y \in \text{women}) (x \text{ comes and } y \text{ leaves}) $$ -
(a) $$ (\forall a \in \mathbb{R}) (\exists x \in \mathbb{R}) (x^2 + a = 0) $$
(b) $$ ((\forall a \in \mathbb{R}) \land (a \lt 0)) (\exists x \in \mathbb{R} )(x^2 + a = 0) $$
(c) $$ \forall x \in \mathbb{R}, \exists p, q \in \mathbb{N} \text{ such that } x = \frac{p}{q} $$
(d) $$ \exists x \in \mathbb{R} \text{ such that } \forall p, q \in \mathbb{N}, x \neq \frac{p^2}{q} $$
(e) $$ ((\forall x \in \mathbb{R}) (\forall p, q \in \mathbb{N}) (x \neq \frac{p}{q})) (\forall y \in \mathbb{N})(\forall m, n \in \mathbb{N})(y\neq \frac{m}{n})(y\gt x)$$
(a) $$(\forall D(x))(M(x))$$
b) $$ (\forall \lnot D(x))(M(x)) $$
c) $$ (\forall M(x)) (D(x)) $$
d) $$ (\exists D(x)) (\lnot M(x)) $$
e) $$ (\exists \lnot D(x)) (M(x)) $$