Question

Translate these statements into English, where the domain for each variable consists of all real numbers.

(a)$\mathbf{\forall }\mathit{x}\mathbf{\exists }\mathit{y}\left(x<y\right)$

(b)$\mathbf{\forall }\mathit{x}\mathbf{\forall }\mathit{y}\left(\left(\left(x\ge 0\right)\wedge \left(y\ge 0\right)\right)\to \left(xy\ge 0\right)\right)$

(c)$\mathbf{\forall }\mathit{x}\mathbf{\forall }\mathit{y}\mathbf{\exists }\mathit{z}\left(xy=z\right)$

Short answer

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.”

Step-by-step solution

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.

Translation of statements into English

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

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.