Express the negations of each of the following statements using quantifier notation as fully as possible, but without using the negation symbol (you may use symbols such as X, #, etc.): a) x is an upper bound for S. (Here, S is a subset of R and x is a real number.) b) is a prime. (Here, is an integer greater than 1.) c) u is rational. (Here, u is a real number.) d) d is the GCD of a and b. (Here, a, b, and d are positive integers.) (Careful!) e) R is a function from A to B. (Here, R is a subset of A X B.) f) f is strictly increasing. (Here, f is a function from R to R.) g) fis injective. (Here, f is a function from A to B.) h) f is surjective. (Here, f is a function from A to B.) i) S is an interval. (Here, S is a subset of R. Use the definition of "interval" in Exer- cise 15.)
Express the negations of each of the following statements using quantifier notation as fully as possible, but without using the negation symbol (you may use symbols such as X, #, etc.): a) x is an upper bound for S. (Here, S is a subset of R and x is a real number.) b) is a prime. (Here, is an integer greater than 1.) c) u is rational. (Here, u is a real number.) d) d is the GCD of a and b. (Here, a, b, and d are positive integers.) (Careful!) e) R is a function from A to B. (Here, R is a subset of A X B.) f) f is strictly increasing. (Here, f is a function from R to R.) g) fis injective. (Here, f is a function from A to B.) h) f is surjective. (Here, f is a function from A to B.) i) S is an interval. (Here, S is a subset of R. Use the definition of "interval" in Exer- cise 15.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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questions d,e,g,h
Transcribed Image Text:Express the negations of each of the following statements using quantifier notation as fully as
possible, but without using the negation symbol (you may use symbols such as , #, etc.):
a) x is an upper bound for S. (Here, S is a subset of R and x is a real number.)
b) is a prime. (Here, is an integer greater than 1.)
c) u is rational. (Here, u is a real number.)
d) d is the GCD of a and b. (Here, a, b, and d are positive integers.) (Careful!)
e) R is a function from A to B. (Here, R is a subset of A X B.)
f) f is strictly increasing. (Here, f is a function from R to R.)
g) fis injective. (Here, f is a function from A to B.)
h) f is surjective. (Here, f is a function from A to B.)
i) S is an interval. (Here, S is a subset of R. Use the definition of "interval" in Exer-
cise 15.)
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