3. Determine the truth value of each of the following quantified statements. If a universally quantified statement is false, provide a counterexample. If an existentially quantified statement is true, provide an example. Otherwise, you don't need to justify. (a) Vx € R (x³ = −1) (b) 3x ER (x³ = −1) (c) 3x ER (x4 = -1) (d) 3x € Z (6x² + x − 1 = 0) (e) Va E R ((-a)² = a²) (f) 3x € R+ (√x = -4) (g) Vx € R+ (√x ≥ 1) (h) Vx € Z+ (√x ≥ 1) (i) Vx € R+ (x² + 3x) (i) 3x € R+ (1-3x > 7) (k) Vn e Z+ (4n is an even number) (1) Vn €Z+ (n5 + 3n² + 2 is an odd number) (m) 3y # Z (y² = 4) (n) 3y Z (y² + 4) (0) Vy Z (y² # 4) (p) 3x((x € Z) ^ (0 < x < 1)) (q) Vx ((x € R) → (√x € R)) (r) 3x((x € Z) ^ (x < 5) ^ (x² > 10)) (s) Vx ((x € R+) → ((x > 0) v (x < 0)))
3. Determine the truth value of each of the following quantified statements. If a universally quantified statement is false, provide a counterexample. If an existentially quantified statement is true, provide an example. Otherwise, you don't need to justify. (a) Vx € R (x³ = −1) (b) 3x ER (x³ = −1) (c) 3x ER (x4 = -1) (d) 3x € Z (6x² + x − 1 = 0) (e) Va E R ((-a)² = a²) (f) 3x € R+ (√x = -4) (g) Vx € R+ (√x ≥ 1) (h) Vx € Z+ (√x ≥ 1) (i) Vx € R+ (x² + 3x) (i) 3x € R+ (1-3x > 7) (k) Vn e Z+ (4n is an even number) (1) Vn €Z+ (n5 + 3n² + 2 is an odd number) (m) 3y # Z (y² = 4) (n) 3y Z (y² + 4) (0) Vy Z (y² # 4) (p) 3x((x € Z) ^ (0 < x < 1)) (q) Vx ((x € R) → (√x € R)) (r) 3x((x € Z) ^ (x < 5) ^ (x² > 10)) (s) Vx ((x € R+) → ((x > 0) v (x < 0)))
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:3. Determine the truth value of each of the following quantified statements. If a universally
quantified statement is false, provide a counterexample. If an existentially quantified
statement is true, provide an example. Otherwise, you don't need to justify.
(a) VxER (x³ = −1)
(b) 3x € R (x³ = −1)
(c) 3x ER (x² = −1)
(d) 3x € Z (6x² + x − 1 = 0)
(e) Va €R ((-a)² = a²)
(f) 3x € R+ (√x = −4)
(g) Vx € R+ (√x ≥ 1)
(h) Vx € Z+ (√√x ≥ 1)
(i) Vx € R+ (x² + 3x)
(j) 3x € R¹ (1− 3x > 7)
(k) Vn € Z† (4n is an even number)
(1) \n €Z+ (n5 + 3n² + 2 is an odd number)
(m) 3y
Z (y² = 4)
(n) 3y
Z (y² + 4)
(0) Vy
Z (y² # 4)
(p) 3x((x € Z) ^ (0 < x < 1))
(q) Vx ((x ¤ R) → (√x ¤ R))
(r) 3x((x € Z) ^ (x < 5) ^ (x² > 10))
(s) Vx ((x € R+) → ((x > 0) v (x < 0)))
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