Determine the truth value of each of the following formulas (the universe of discourse for each statement is given in parentheses).

 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