Short Answer

Translate these statements into English, where the domain for each variable consists of all real numbers.
(a) $\forall x \exists y (x < y)$
(b) $\forall x \forall y (((x \geq 0) \land (y \geq 0)) \to (x y \geq 0))$
(c) $\forall x \forall y \exists z (x y = z)$

For expressing the given statements in English, use the significance of quantifiers. Here, the quantifier “ $\forall$ ” indicates “All” whereas the quantifier “ $\exists$ ” represents “Some” or “There exists.”

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of Quantifier

Quantifiers are terms that correspond to quantities such as "some" or "all" and indicate the number of items for which a certain proposition is true.

Step 2: Translation of statements into English

(a) $\forall x \exists y (x < y)$

This indicates that for every real number x there exists a real number y such that x is less than y.

(b) $\forall x \forall y (((x \geq 0) \land (y \geq 0)) \to (x y \geq 0))$

This indicates that for every real number x and real number y if x and y are both non-negative, then their product is non-negative.

This indicates that for every real number x and real number y there exists a real number z such that $x y = z$ .

Therefore, the given statements have been expressed in English.

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Short Answer

Translate these statements into English, where the domain for each variable consists of all real numbers.
(a) $\forall x \exists y (x < y)$
(b) $\forall x \forall y (((x \geq 0) \land (y \geq 0)) \to (x y \geq 0))$
(c) $\forall x \forall y \exists z (x y = z)$

Step by Step Solution

Step 1: Definition of Quantifier

Step 2: Translation of statements into English

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Discrete Mathematics and its Applications

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Translate these statements into English, where the domain for each variable consists of all real numbers.(a) ∀x∃y(x<y)(b) ∀x∀y(x≥0∧y≥0→xy≥0)(c) ∀x∀y∃z(xy=z)

Step by Step Solution

Step 1: Definition of Quantifier

Step 2: Translation of statements into English

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Translate these statements into English, where the domain for each variable consists of all real numbers.
(a) $\forall x \exists y (x < y)$
(b) $\forall x \forall y (((x \geq 0) \land (y \geq 0)) \to (x y \geq 0))$
(c) $\forall x \forall y \exists z (x y = z)$