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Found in: Page 64

### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

# Translate these statements into English, where the domain for each variable consists of all real numbers.(a) ${\mathbf{\forall }}{\mathbit{x}}{\mathbf{\exists }}{\mathbit{y}}\left(x(b) ${\mathbf{\forall }}{\mathbit{x}}{\mathbf{\forall }}{\mathbit{y}}\left(\left(\left(x\ge 0\right)\wedge \left(y\ge 0\right)\right)\to \left(xy\ge 0\right)\right)$(c) ${\mathbf{\forall }}{\mathbit{x}}{\mathbf{\forall }}{\mathbit{y}}{\mathbf{\exists }}{\mathbit{z}}\left(xy=z\right)$

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.”

See the step by step solution

## 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\left(x

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

(b) $\forall x\forall y\left(\left(\left(x\ge 0\right)\wedge \left(y\ge 0\right)\right)\to \left(xy\ge 0\right)\right)$

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.

(c) $\forall x\forall y\exists z\left(xy=z\right)$

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

Therefore, the given statements have been expressed in English.