Question

Express each of these mathematical statements using predicates, quantifiers, logical connectives, and mathematical operators.

a) The product of two negative real numbers is positive.

b) The difference of a real number and itself is zero.

c) Every positive real number has exactly two square roots.

d) A negative real number does not have a square root that is a real number. e) Every non-zero real number has a unique reciprocal. 

Answer 1

$a)\,\forall x\in\mathbb{R}\; \forall y\in\mathbb{R} \: (x<0\: \land \: y<0) \implies (x\cdot y>0)$

$b) \,\forall x\in\mathbb{R} \: (x-x=0)$

$c) \, \forall x \in\mathbb{R} \: (x>0) \implies \exist y_1\in\mathbb{R}\; \exists y_2\in\mathbb{R} \: (y_1\neq y_2)\land\\ \land (x=y_1^2=y_2^2) \land \\ \land (\forall y \in\mathbb{R} \: (y^2=x)\implies (y=y_1 \lor y=y_2))$

$d)\,\forall x\in\mathbb{R}\; (x<0) \implies \forall y\in\mathbb{R} \: (y^2\neq x)$

$e) \,\forall x \in\mathbb{R} \; (x\neq 0) \implies (\exists y \in\mathbb{R} \; (x\cdot y=1) \land \\ \land (\forall z\in\mathbb{R} \; (x\cdot z=1) \implies (z=y)))$