Question

Translate each of these quantifications into English language and determine its truth

value.

1. ∃x∈R (x3 = −1)

2. ∀x∈Z (x2 ∈ Z)

Answer 1

1. There exists a real number $x$ that satisfies equation: $x^3=-1$.

The statement is true. This number is $x=-1$.

2. For all integers $x$ a number $x^2$ is also integer.

The statement is true. Any product of integers is also an integer. In particular, for any integer $x$ a number $x^2$ is also an integer.