Rewrite each of the following statements using the quantifiers “for all” (or “for every”) and “there exists” as appropriate.
(a) [BB] Not all continuous functions are differentiable.
(b) For real x, 2x is never negative.
(c) [BB] There is no largest real number.
(d) There are infinitely many primes.
(e) [BB] Every positive integer is the product of primes.
(f) All positive real numbers have real square roots.
(g) [BB] There is no smallest integer.
(h) There is no smallest positive real number.
(i) [BB] Not every polynomial has a real root.
(j) Between every two (different) real numbers lies a rational number.
(k) Every polynomial of degree 3 has a real root.
(l) Not every nonzero matrix is invertible.
(m) Some real numbers are nonnegative.
(n) An integer cannot be both even and odd.
(o) If a, b, and c are nonzero integers, a3 + b3 ≠ c3.
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