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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 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 E Z+ (√x ≥ 1) (i) Vx € R+ (x² + 3x) (j) 3x ER+ (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 EZ) ^ (0 < x < 1)) A (q) Vx ((x € R) → (√x ER)) (r) 3x ((x € Z) ^ (x < 5) ^ (x² > 10)) (s) Vx ((x ER+) → ((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) VxER (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 E Z+ (√x ≥ 1) (i) Vx € R+ (x² + 3x) (j) 3x ER+ (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 EZ) ^ (0 < x < 1)) A (q) Vx ((x € R) → (√x ER)) (r) 3x ((x € Z) ^ (x < 5) ^ (x² > 10)) (s) Vx ((x ER+) → ((x > 0) v (x < 0)))

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 35E
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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 ER (x³ = −1) (c) ExER (x¹ = −1) (d) 3x € Z (6x² + x − 1 = 0) (e) Va ER ((-a)² = a²) (f) 3x € R+ (√x = −4) (g) Vx € R¹ (√x ≥ 1) (h) Vx € Z+ (√x ≥ 1) (1) Vx E R+ (x² + 3x) (j) 3x € R+ (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) → (√XER)) E (r) 3x ((x € Z) ^ (x < 5) ^ (x² > 10)) (s) Vx ((x € R+) → ((x > 0) v (x < 0))) E
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 ER (x³ = −1) (c) ExER (x¹ = −1) (d) 3x € Z (6x² + x − 1 = 0) (e) Va ER ((-a)² = a²) (f) 3x € R+ (√x = −4) (g) Vx € R¹ (√x ≥ 1) (h) Vx € Z+ (√x ≥ 1) (1) Vx E R+ (x² + 3x) (j) 3x € R+ (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) → (√XER)) E (r) 3x ((x € Z) ^ (x < 5) ^ (x² > 10)) (s) Vx ((x € R+) → ((x > 0) v (x < 0))) E
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