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Quantifications and Arguments. 1. Determine the truth value of the following quantifications. Include their negations. a. Some mammals have an aquatic habitat. b. At least one pine tree can produce a flower. c. All real numbers have numbers have positive squares.

Quantifications and Arguments. 1. Determine the truth value of the following quantifications. Include their negations. a. Some mammals have an aquatic habitat. b. At least one pine tree can produce a flower. c. All real numbers have numbers have positive squares.

Advanced Engineering Mathematics
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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Quantifications and Arguments. 1. Determine the truth value of the following quantifications. Include their negations. a. Some mammals have an aquatic habitat. b. At least one pine tree can produce a flower. c. All real numbers have numbers have positive squares. d. Each counting number is greater than a negative integer. e. No planet in the solar system is surrounded by rings of ice, rock or dust. NOTE: Please use the photo attached as reference
V XE U, p(x). 3 x€ U, p(x). I. True να ευ p(a) is true 3 a€ U p(a) is true 1. Each scorpion is an arachnid. U = { scorpions} a(x) : x is an arachnid. "VxE U, a(x)," is true. 2. No grass can stand as high as 6 ft. U = {grass} h(x) : x can stand as high as 6 ft. "VxEU, ~h(x)," is false. 3. Every integer is a rational number. U= { integers } = Z q(x): x is a rational number. "VxEU, q(x)," is true. "VxEZ, q(x)," is true. 4. All real numbers are rational numbers. V= { real numbers } = R q(x): x is a rational number. "V XE U, q(x)," is false. "VxE R, q(x)," is false. 5. Some volcanoes are located under the sea. U = { volcanoes } I(x) : x is located under the sea. "3 XE U, I(x)," is true. 6. There is an integer that satisfies x² = 1. V = { integers } = Z e(x) : x satisfies x² = 1. "3 xE U, e(x)," is true. "3 xE Z, e(x)," is true. False 3 a€ U p(a) is false Vae V p(a) is false
Transcribed Image Text:V XE U, p(x). 3 x€ U, p(x). I. True να ευ p(a) is true 3 a€ U p(a) is true 1. Each scorpion is an arachnid. U = { scorpions} a(x) : x is an arachnid. "VxE U, a(x)," is true. 2. No grass can stand as high as 6 ft. U = {grass} h(x) : x can stand as high as 6 ft. "VxEU, ~h(x)," is false. 3. Every integer is a rational number. U= { integers } = Z q(x): x is a rational number. "VxEU, q(x)," is true. "VxEZ, q(x)," is true. 4. All real numbers are rational numbers. V= { real numbers } = R q(x): x is a rational number. "V XE U, q(x)," is false. "VxE R, q(x)," is false. 5. Some volcanoes are located under the sea. U = { volcanoes } I(x) : x is located under the sea. "3 XE U, I(x)," is true. 6. There is an integer that satisfies x² = 1. V = { integers } = Z e(x) : x satisfies x² = 1. "3 xE U, e(x)," is true. "3 xE Z, e(x)," is true. False 3 a€ U p(a) is false Vae V p(a) is false
Quantifications area statements of properties that are satisfied either by the entire set or some of its elements. Universal Quantification Existential Quantification V 3 for all, for every, for each, for any for some, at least one, there exists, there is VXEU, p(x). 3x U, p(x).
Transcribed Image Text:Quantifications area statements of properties that are satisfied either by the entire set or some of its elements. Universal Quantification Existential Quantification V 3 for all, for every, for each, for any for some, at least one, there exists, there is VXEU, p(x). 3x U, p(x).
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