A Transition to Advanced Mathematics
8th Edition
ISBN: 9781285463261
Author: Douglas Smith, Maurice Eggen, Richard St. Andre
Publisher: Cengage Learning
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Textbook Question
Chapter 1.5, Problem 1E
Analyze the logical form of each of the following statements, and construct just the outline of a proof by the given method. Do not provide any details of the proof.
- Outline a proof by contraposition that if
- Outline a proof by contraposition that if B is a nonsingular matrix, then the determinant of B is not zero.
- Outline a proof by contradiction that the set of natural numbers is not finite.
- Outline a proof by contradiction that if x is a nonzero real number, the multiplicative inverse of x is unique.
- Outline a two-part proof that the inverse of the function f from A to B is a function from B to A if and only if f is one-to-one and fis onto B.
- Outline a two-part proof that a subset A of the real numbers is compact if and only if A is closed and bounded.
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Students have asked these similar questions
2. Consider the ODE
u' = ƒ (u) = u² + r
where r is a parameter that can take the values r = −1, −0.5, -0.1, 0.1. For each value of r:
(a) Sketch ƒ(u) = u² + r and determine the equilibrium points.
(b) Draw the phase line.
(d) Determine the stability of the equilibrium points.
(d) Plot the direction field and some sample solutions,i.e., u(t)
(e) Describe how location of the equilibrium points and their stability change as you increase the
parameter r.
(f) Using the matlab program phaseline.m generate a solution for each value of r and the initial
condition u(0) = 0.9. Print and turn in your result for r = −1. Do not forget to add a figure caption.
(g) In the matlab program phaseline.m set the initial condition to u(0) = 1.1 and simulate the ode
over the time interval t = [0, 10] for different values of r. What happens? Why? You do not need to
turn in a plot for (g), just describe what happens.
True or False and why
10
5
Obtain by multiplying matrices the composite coordinate transformation of two transformations, first
x' = (x + y√√2+2)/2
y' =
z'
(x√√2-2√2)/2
z = (-x+y√√2-2)/2
followed by
x"
=
(x'√√2+z'√√2)/2
y" = (-x'y'√√2+2')/2
z" = (x'y'√√2-2')/2.
Chapter 1 Solutions
A Transition to Advanced Mathematics
Ch. 1.1 - Which of the following are propositions? Give the...Ch. 1.1 - For each pair of statements, determine whether the...Ch. 1.1 - Make a truth table for each of the following...Ch. 1.1 - If P, Q, and R are true while S and K are false,...Ch. 1.1 - Use truth tables to verify each part of Theorem...Ch. 1.1 - Which of the following pairs of propositional...Ch. 1.1 - Determine the propositional form and truth value...Ch. 1.1 - Suppose P, Q, and R are propositional forms....Ch. 1.1 - Suppose P, Q, S, and R are propositional forms, P...Ch. 1.1 - Use a truth table to determine whether each of the...
Ch. 1.1 - Give a useful denial of each statement. Assume...Ch. 1.1 - Restore parentheses to these abbreviated...Ch. 1.1 - Other logical connectives between two propositions...Ch. 1.1 - Other logical connectives between two propositions...Ch. 1.2 - Identify the antecedent and the consequent for...Ch. 1.2 - Prob. 2ECh. 1.2 - What can be said about the truth value of Q when...Ch. 1.2 - Identify the antecedent and the consequent for...Ch. 1.2 - Which of the following conditional sentences are...Ch. 1.2 - Which of the following are true? Assume that x and...Ch. 1.2 - Make truth tables for these propositional forms....Ch. 1.2 - Prove Theorem 1.2.2 by constructing truth tables...Ch. 1.2 - Determine whether each statement qualifies as a...Ch. 1.2 - Prob. 10ECh. 1.2 - Dictionaries indicate that the conditional meaning...Ch. 1.2 - Show that the following pairs of statements are...Ch. 1.2 - Prob. 13ECh. 1.2 - Give, if possible, an example of a false...Ch. 1.2 - Give the converse and contrapositive of each...Ch. 1.2 - Prob. 16ECh. 1.2 - The inverse, or opposite, of the conditional...Ch. 1.3 - Translate the following English sentences into...Ch. 1.3 - For each of the propositions in Exercise 1, write...Ch. 1.3 - Translate these definitions from the Appendix into...Ch. 1.3 - Prob. 4ECh. 1.3 - The sentence “People dislike taxes” might be...Ch. 1.3 - Let T={17},U={6},V={24} , and W={2,3,7,26} . In...Ch. 1.3 - (a) Complete the following proof of Theorem...Ch. 1.3 - Which of the following are true? The universe for...Ch. 1.3 - Give an English translation for each. The universe...Ch. 1.3 - Which of the following are true in the universe of...Ch. 1.3 - Let A(x) be an open sentence with variable x. (a)...Ch. 1.3 - Suppose the polynomials anxn+an1xn1+...+a0 and...Ch. 1.3 - Which of the following are denials of (!x)P(x) ?...Ch. 1.3 - Riddle: What is the English translation of the...Ch. 1.4 - Analyze the logical form of each of the following...Ch. 1.4 - A theorem of linear algebra states that if A andB...Ch. 1.4 - Verify that [(BM)L(ML)]B is a tautology. See the...Ch. 1.4 - These facts have been established at a crime...Ch. 1.4 - Prob. 5ECh. 1.4 - Let a and b be real numbers. Prove that (a)...Ch. 1.4 - Suppose a, b, c, and d are integers. Prove that...Ch. 1.4 - Give two proofs that if n is a natural number,...Ch. 1.4 - Let a, b, and c be integers and x, y, and z be...Ch. 1.4 - Recall that except for degenerate cases, the graph...Ch. 1.4 - Exercises throughout the text with this title ask...Ch. 1.5 - Analyze the logical form of each of the following...Ch. 1.5 - A theorem of linear algebra states that if A andB...Ch. 1.5 - Let x, y, and z be integers. Write a proof by...Ch. 1.5 - Write a proof by contraposition to show that for...Ch. 1.5 - A circle has center (2,4) . (a) Prove that (1,5)...Ch. 1.5 - Suppose a and b are positive integers. Write a...Ch. 1.5 - Prob. 7ECh. 1.5 - Prob. 8ECh. 1.5 - Prove by contradiction that if n is a natural...Ch. 1.5 - Prove that 5 is not a rational number.Ch. 1.5 - Three real numbers, x, y, and z, are chosen...Ch. 1.5 - Assign a grade of A (correct), C (partially...Ch. 1.6 - Prove that (a) there exist integers m and n such...Ch. 1.6 - Prove that for all integers a, b, and c, If...Ch. 1.6 - Prove that if every even natural number greater...Ch. 1.6 - Provide either a proof or a counterexample for...Ch. 1.6 - (a) Prove that the natural number x is prime if...Ch. 1.6 - Prove that (a) for every natural number n, 1n1 ....Ch. 1.6 - Starting at 9 a.m. on Monday, a hiker walked at a...Ch. 1.6 - Show by example that each of the following...Ch. 1.6 - Assign a grade of A (correct), C (partially...Ch. 1.7 - (a) Let a be a negative real number. Prove that if...Ch. 1.7 - Prob. 2ECh. 1.7 - Prove that (a) 5n2+3n+4 is even, for all integers...Ch. 1.7 - Prob. 4ECh. 1.7 - Prove that (a) if x + y is irrational, then either...Ch. 1.7 - Prob. 6ECh. 1.7 - Prob. 7ECh. 1.7 - Prob. 8ECh. 1.7 - Prob. 9ECh. 1.7 - Prob. 10ECh. 1.7 - Assign a grade of A (correct), C (partially...Ch. 1.8 - For each given pair a, b of integers, find the...Ch. 1.8 - Prob. 2ECh. 1.8 - Let a and b be integers, a0 , and ab . Prove that...Ch. 1.8 - Prob. 4ECh. 1.8 - Prob. 5ECh. 1.8 - Prob. 6ECh. 1.8 - Prob. 7ECh. 1.8 - Prob. 8ECh. 1.8 - Prove that for every prime p and for all natural...Ch. 1.8 - Let q be a natural number greater than 1 with the...Ch. 1.8 - Prob. 11ECh. 1.8 - Prob. 12ECh. 1.8 - Let a and b be nonzero integers that are...Ch. 1.8 - Let a and b be nonzero integers and d=gcd(a,b) ....Ch. 1.8 - Let a and b be nonzero integers and c be an...Ch. 1.8 - Prob. 16ECh. 1.8 - Prob. 17ECh. 1.8 - Let a and b be integers, and let m=lcm(a,b) . Use...Ch. 1.8 - The greatest common divisor of positive integers a...Ch. 1.8 - Prob. 20ECh. 1.8 - Prob. 21E
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