Find the truth set of each of these predicates where the domain is the set of integers.

a. p (x) : x3 ≥ 1

b. q (x) : x2 = 2

c. r (x) : x < x2

a. p (x) : x^3 ≥ 1

x^3 ≥ 1 = > x ≥ 1 = > all the integers equal or greater than 1 = x ∈ Z = {1, 2, 3, 4, 5, 6, ... }

That is the same that the natural numbers except 0 = N - {0}.

b. q (x) : x2 = 2

x^2 = 2 = > x = + / - √2, which is not an integer, so the truth set is the empty set = { } = ∅

c. r (x) : x < x2

x x^2 - x > 0

=> x (x - 1) > 0

=>

1) x > 0 and x - 1 > 0

=> x > 0 and x > 1

=> x > 1

2) x < 0 and x - 1 < 0

=> x < 0 and x x < 0

=> the solution is the union of the two sets: x > 1 ∪ x < 0

That is all the integers except 0 and 1.

=> the truth set is x ∈ Z = Z - { 0,1}