Why Read This Book? - Index of

Solution
1. Everyone in this class did not cheat on the final exam.
1.4 Negations of Statements 39
2. For every natural number x, there exists a natural number y such that x>y.
3. For all real numbers M, there exists x ∈ S such that x>M.
4. There exists a ∈ A and ɛ>0 such that, for all δ>0, there exists an x satisfying
0 < |x − a| 1 ∧ a ≤ 10
(b) a ≥ 120 ∨ a ≤ 80
(c) (∀x)(x 2 − 4x + 3 = 0 → x ≥ 1)
(d) (∃x)(x ≥ 2 ∧ x 2 − 3x = 2)
(e) (∀a ∈ R)(∃b ∈ R)(a + b = 0)
(f) (∃!x)(P(x))
(g) (∀a ∈ R)(∃!b ∈ R)(a + b = 0)
(h) (∃m ∈ A)(∀x ∈ A)(x ≤ m)
(i) (∀x)(x ∈ A → x ∈ B)
(j) (∃x)(x ∈ A ∧ x ∈ B)
(k) There is an element of set A that is less than every element of set B.
(l) Every element of A is greater than every element of B.
(m) There exists an element of the set A that exceeds m.
(n) The graph of every function in F intersects the x-axis at least once.
(o) There is a function in F whose graph intersects the x-axis.
(p) No function in F has a graph that intersects the x-axis.
(q) There is a function in F whose graph does not intersect the x-axis.
(r) The equation x 3 = 10 has a real number solution.
(s) There exist M1,M2 ∈ R such that M1 ≤ x ≤ M2 for all x ∈ S.
(t) There exists a unique x ∈ A.
(u) Every real number has a multiplicative inverse.
(v) Every integer is either even or odd.