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
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ISBN:9780470458365
Author:Erwin Kreyszig
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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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