Question

Which disjunction is always true for any real number x?

A number, x, is less than 0 or greater than 1.
A number, x, is less than 0 or greater than 0.
A number, x, is greater than 5 or less than 10.
A number, x, is greater than 5 or less than 1.

Answer 1

The third one: "x > 5 or x < 10."

Reason: For any real x either x < 10 (then the disjunction is true) or x ≥ 10 (then x > 5 and the disjunction is true). Equivalently, the negation would require x ≤ 5 and x ≥ 10 simultaneously, which is impossible.