Answer to Question #308966 in Discrete Mathematics for Ahmad

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.

a) ∃x ∀y(x + y = y)

b) ∀x ∀y (((x ≥ 0) ∧ (y < 0)) → (x − y > 0))

c) ∃x ∃y(((x ≤ 0) ∧ (y ≤ 0)) ∧ (x − y > 0))

d) ∀x ∀y((x ≠ 0) ∧ (y ≠ 0) ↔ (xy ≠ 0))

Answer 1

a) There exists an element x such that if it is added with any number y it does not change this number.

b) For any numbers x and y if x is greater than or equal to 0 and y is less than 0, their difference will be greater than 0.

c) There exist numbers x and y such that x and y are both less than or equal to 0 and their difference is greater than 0.

d) For any numbers x and y they both are not equal to 0 if and only if their product is not euqal to 0.