Question

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

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

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

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

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

This indicates that there exists a real number x such that for every real number y the product of x and y is equal to y.

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

This indicates that for every real number x and every real number y if x is non-negative and y is negative, then their difference is $x-y$positive.

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

This indicates that for every real number x and every real number y there exists a real number z such that the sum of y and z equals x.

Therefore, the given statements have been expressed in English.