Chapter 0.1, Problem 7E

Rewrite each of the following statements using the quantifiers “for all”(or “for every”) and “there exists” as appropriate.

1. Not all continuous functions are differentiable.

2. For real $x$, $2^x$ is never negative.

3. There is no largest real number.

4. There are infinitely many primes.

5. Every positive integer is the product of primes.

6. All positive real numbers have real square roots.

7. There is no smallest integer.

8. There is smallest positive real number.

9. Not every polynomial has a real root.

10. Between every two (different) real numbers lies a rational number.

11. Every polynomial of degree 3 has a real root.

12. Not every nonzero matrix is invertible.

13. Some real numbers are nonnegative.

14. An integer cannot be both even and odd.

15. If $a,b,\text{and}c$ are nonzero integers, $a^3+b^3\ne c^3$.