Question

Translate each of these nested quantifications into an English statement that expresses a mathematical fact. The domain in each case consists of all real numbers.
\[\begin{array}{l}a)\,\exists x\forall y(xy = y)\\b)\,\forall x\forall y(((x < 0) \wedge (y < 0)) \to (xy > 0))\\c)\,\exists x\exists y((({x^2} > y) \wedge (x < y))\\d)\,\forall x\forall y\exists z(x + y = z)\end{array}\]

Short answer

The nested quantifications into English statement that expresses a mathematical fact can be translated.

Step-by-step solution

Translation of the nested quantification for a

a) There exist a real number x as \(\exists x\) and every number y as \(\forall y.\) The product of number x to y and will result in real number y, xy=y. It can be translated as “Subtracting every negative real number from every positive real number will results in a positive real number”.

Translation of the nested quantification for b

b) For every real number x and y as \(\forall x\forall y.\) The negative real number x and positive real number y as \((x \ge 0) \wedge (y < 0).\)The product of x and y will result in a positive number as xy>0. It can be translated as “The product of two negative real numbers is always a positive real number”.

Translation of the nested quantification for c

c) For every real number x and y as \(\exists x\exists y.\)It exceeds y but x is less than y \((({x^2} > y) \wedge (x < y)) \Rightarrow {x^2}.\) It can be translated as “There exists real numbers x and y such that x² exceeds y but x is less than y”.

Translation of the nested quantification for d

d) For every real number x and y as \(\forall x\forall y.\)There exist z as\(\exists z.\)The sum of x and y is a real number as x+y=z. It can be translated as “There exist a real number which is equal to the sum of every two real numbers”.