Question: Determine the truth value of each of the following formulas (the universe of discourse for each statement is given in parentheses).
(a) (∀x)(x + x ≥ x). (ℝ)
(b) (∀x)(x + x ≥ x). (ℕ)
(c) (∃ x) (2x = x). (ℕ)
(d) (∃ x) (2x = x). (w)
Solution:
a) (∀x)(x + x ≥ x). (ℝ)
False
Counter Example: x = -2
b) (∀x)(x + x ≥
x). (ℕ)
True
c) (∃ x) (2x =
x). (ℕ)
False
There is no such number
d) (∃ x) (2x =
x). (w)
True
Example: x = 0 and 2x = 0
