1. The problem asks to translate logical statements involving quantifiers and inequalities into English, assuming the domain of all variables is all real numbers. 2. Statement a) is \(\forall x \exists y (x < y)\). This means "For every real number \(x\), there exists a real number \(y\) such that \(x < y\)." In simpler terms, no matter what number you pick, you can find a bigger number. 3. Statement b) is \(\forall x \forall y (((x \geq 0) \wedge (y \geq 0)) \to (xy \geq 0))\). This means "For all real numbers \(x\) and \(y\), if \(x\) and \(y\) are both greater than or equal to zero, then their product \(xy\) is also greater than or equal to zero." This expresses that the product of two nonnegative numbers is nonnegative. 4. Statement c) is \(\forall x \forall y \exists z (xy = z)\). This means "For every real number \(x\) and every real number \(y\), there exists a real number \(z\) such that \(z\) equals the product of \(x\) and \(y\)." This states that the product of any two real numbers is also a real number. Final translations: a) For every real number, there is a larger real number. b) The product of any two nonnegative real numbers is nonnegative. c) The product of any two real numbers is a real number.