Translate "∀x∃y(x < y)" in English, considering the domain as real numbers for both variables.
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Step 1
Step 1: The symbol "∀" means "for all" and the symbol "∃" means "there exists". Show more…
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Translate these statements into English, where the domain for each variable consists of all real numbers. a) $\quad \forall x \exists y(x<y)$ b) $\forall x \forall y(((x \geq 0) \wedge(y \geq 0)) \rightarrow(x y \geq 0))$ c) $\forall x \forall y \exists z(x y=z)$
The Foundations: Logic and Proofs
Nested Quantifiers
Translate each of these nested quantifications into an English statement that expresses a mathematical fact. The domain in each case consists of all real numbers. a) $\exists x \forall y(x y=y)$ b) $\forall x \forall y(((x<0) \wedge(y<0)) \rightarrow(x y>0))$ c) $\exists x \exists y\left(\left(x^{2}>y\right) \wedge(x<y)\right)$ d) $\forall x \forall y \exists z(x+y=z)$
Translate each of these nested quantifications into an English statement that expresses a mathematical fact. The domain in each case consists of all real numbers. a) $\exists x \forall y(x+y=y)$ b) $\forall x \forall y(((x \geq 0) \wedge(y<0)) \rightarrow(x-y>0))$ c) $\exists x \exists y(((x \leq 0) \wedge(y \leq 0)) \wedge(x-y>0))$ d) $\forall x \forall y((x \neq 0) \wedge(y \neq 0) \leftrightarrow(x y \neq 0))$
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