Discrete Mathematics With Applications
5th Edition
ISBN: 9781337694193
Author: EPP, Susanna S.
Publisher: Cengage Learning,
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Textbook Question
Chapter 3.4, Problem 7ES
Some of the arguments in 7-18 are valid by universal modus ponens or universal modus tollens; other are invalid and exhibit the converse or the inverse error. State which are valid and which are invalid. Justify your answers.
All healthy people eat an apple a day.
Keisha eats an apple a day.
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This question builds on an earlier problem. The randomized numbers may have changed, but have your work for the previous problem available to help with this one.
A 4-centimeter rod is attached at one end to a point A rotating counterclockwise on a wheel of radius 2 cm. The other end B is free to move back and forth along a horizontal bar that goes through the center of the wheel. At time t=0 the rod is situated as in the diagram at the left below. The
wheel rotates counterclockwise at 1.5 rev/sec. At some point, the rod will be tangent to the circle as shown in the third picture.
A
B
A
B
at some instant, the piston will be tangent to the circle
(a) Express the x and y coordinates of point A as functions of t:
x= 2 cos(3πt)
and y= 2 sin(3t)
(b) Write a formula for the slope of the tangent line to the circle at the point A at time t seconds:
-cot(3πt)
sin(3лt)
(c) Express the x-coordinate of the right end of the rod at point B as a function of t: 2 cos(3πt) +411-
4
-2 sin (3лt)
(d)…
5. [-/1 Points]
DETAILS
MY NOTES
SESSCALCET2 6.5.AE.003.
y
y= ex²
0
Video Example
x
EXAMPLE 3
(a) Use the Midpoint Rule with n = 10 to approximate the integral
कर
L'ex²
dx.
(b) Give an upper bound for the error involved in this approximation.
SOLUTION
8+2
1
L'ex² d
(a) Since a = 0, b = 1, and n = 10, the Midpoint Rule gives the following. (Round your answer to six decimal places.)
dx Ax[f(0.05) + f(0.15) + ... + f(0.85) + f(0.95)]
0.1 [0.0025 +0.0225
+
+ e0.0625 + 0.1225
e0.3025 + e0.4225
+ e0.2025
+
+ e0.5625 €0.7225 +0.9025]
The figure illustrates this approximation.
(b) Since f(x) = ex², we have f'(x)
=
0 ≤ f'(x) =
< 6e.
ASK YOUR TEACHER
and f'(x) =
Also, since 0 ≤ x ≤ 1 we have x² ≤
and so
Taking K = 6e, a = 0, b = 1, and n = 10 in the error estimate, we see that an upper bound for the error is as follows. (Round your final
answer to five decimal places.)
6e(1)3
e
24(
=
≈
1. Consider the following preference ballots:
Number of voters
Rankings 6 5 4 2
1st choice A DCB
DC
2nd choice B B D
3rd choice DCBD
4th choice CA
AAA
For each of the four voting systems we have studied, determine who would win the election
in each case. (Remember: For plurality with runoff, all but the top two vote-getters are
simultaneously eliminated at the end of round 1.)
Chapter 3 Solutions
Discrete Mathematics With Applications
Ch. 3.1 - If P(x) is a predicate with domain D, the truth...Ch. 3.1 - Some ways to express the symbol in words are .Ch. 3.1 - Some ways to express the symbol in words are .Ch. 3.1 - A statement of from xD , Q(x) is true if, and only...Ch. 3.1 - A statement of the form xD such that Q(x) is true:...Ch. 3.1 - A menagerie consists of seven brown dogs, two...Ch. 3.1 - Indicate which of the following statements are...Ch. 3.1 - Let R(m,n) be the predicate “If m is a factor if...Ch. 3.1 - Let Q(x,y) be the predicate “If xy then x2y2 ”...Ch. 3.1 - Find the truth set of each predicate. Predicate:...
Ch. 3.1 - Let B(x) be “ 10x10 .” Find the truth set of B(x)...Ch. 3.1 - Let S be the set of all strings of length 3...Ch. 3.1 - Let T be the set of all strings of length 3...Ch. 3.1 - Find counterexamples to show that the statements...Ch. 3.1 - Find counterexamples to show that the statements...Ch. 3.1 - Find counterexamples to show that the statements...Ch. 3.1 - Find counterexamples to show that the statements...Ch. 3.1 - Consider the following statement: basketball...Ch. 3.1 - Consider the following statement: xR such that...Ch. 3.1 - Rewrite the following statements informally in at...Ch. 3.1 - Rewrite each of the following statements in the...Ch. 3.1 - Rewrite each of the following in the form “ _____...Ch. 3.1 - Let D be the sat of all students at your school,...Ch. 3.1 - Consider the following statement: integer n, if...Ch. 3.1 - Rewrite the following statement informally in at...Ch. 3.1 - Prob. 21ESCh. 3.1 - Rewrite each of the following statements in the...Ch. 3.1 - Rewrite each of the following statements in the...Ch. 3.1 - Rewrite the following statements in the two forms...Ch. 3.1 - The statement “The square of any rational number...Ch. 3.1 - Consider the statement “All integers are rational...Ch. 3.1 - Refer to the picture of Tarski’s world given in...Ch. 3.1 - In 28-30, rewrite each statement without using...Ch. 3.1 - Let the domain of x be the set of geometric...Ch. 3.1 - Let the domain of x be Z, the set of integers, and...Ch. 3.1 - In any mathematics or computer science text other...Ch. 3.1 - Let R be the domain of the predicate variable x....Ch. 3.1 - Prob. 33ESCh. 3.2 - A negation for “All R have property S” is “There...Ch. 3.2 - A negation for “Some R have property S” is...Ch. 3.2 - A negation for “For every x, if x has property P...Ch. 3.2 - The converse of “For every x, if x has property P...Ch. 3.2 - The contrapositive of “For every x, if x has...Ch. 3.2 - The inverse of “For every x, if x has property P...Ch. 3.2 - Which of the following is a negation for “All...Ch. 3.2 - Which of the following is a negation for “All dogs...Ch. 3.2 - Write a formula negation for each of the following...Ch. 3.2 - Write an informal negation for each of the...Ch. 3.2 - Write a negation for each of the following...Ch. 3.2 - Write a negation for each statement in 6 and 7....Ch. 3.2 - Write a negation for each statement in 6 and 7....Ch. 3.2 - Consider the statement “There are no simple...Ch. 3.2 - Write negation for each statement in 9 and 10. ...Ch. 3.2 - Write a negation for each statements in 9 and 10. ...Ch. 3.2 - In each of 11-14 determine whether the proposed...Ch. 3.2 - In each of 11-14 determine whether the proposed...Ch. 3.2 - Prob. 13ESCh. 3.2 - Prob. 14ESCh. 3.2 - Prob. 15ESCh. 3.2 - In 16-23, write a negation for each statement. ...Ch. 3.2 - In 16-23, write a negation for each statement. ...Ch. 3.2 - Prob. 18ESCh. 3.2 - In 16-23, write a negation for each statement. nZ...Ch. 3.2 - Prob. 20ESCh. 3.2 - Prob. 21ESCh. 3.2 - In 16-23, write a negation for each statement. If...Ch. 3.2 - In 16-23, write a negation for each statement. If...Ch. 3.2 - Prob. 24ESCh. 3.2 - Prob. 25ESCh. 3.2 - In 26-33, for each statement in the referenced...Ch. 3.2 - In 26-33, for each statement in the referenced...Ch. 3.2 - Prob. 28ESCh. 3.2 - In 26-33, for each statement in the referenced...Ch. 3.2 - Prob. 30ESCh. 3.2 - In 26-33, for each statement in the referenced...Ch. 3.2 - In 26-33, for each statement in the referenced...Ch. 3.2 - In 26-33, for each statement in the referenced...Ch. 3.2 - Prob. 34ESCh. 3.2 - Give an example to show that a universal condition...Ch. 3.2 - If P(x) is a predicate and the domain of x is the...Ch. 3.2 - Consider the following sequence of digits: 0204. A...Ch. 3.2 - True or false? All occurrences of the letter u in...Ch. 3.2 - Prob. 39ESCh. 3.2 - Rewrite each statement of 39-44 if-then form....Ch. 3.2 - Rewrite each statement of 39-44 if-then form....Ch. 3.2 - Rewrite each statement of 39-44 if-then form....Ch. 3.2 - Rewrite each statement of 39-44 in if-then form. A...Ch. 3.2 - Rewrite each statement of 39-44 in if-then form. A...Ch. 3.2 - Prob. 45ESCh. 3.2 - Use the facts that the negation of a STATEMENT IS...Ch. 3.2 - Prob. 47ESCh. 3.2 - Use the facts that the negation of STATEMENT IS A...Ch. 3.2 - The computer scientist Richard Conway and David...Ch. 3.2 - A frequent-flyer club brochure stares, “you may...Ch. 3.3 - To establish the truth of a statement of the form...Ch. 3.3 - Prob. 2TYCh. 3.3 - Prob. 3TYCh. 3.3 - Consider the statement “ x such that y , P(x,y), a...Ch. 3.3 - Prob. 5TYCh. 3.3 - Prob. 1ESCh. 3.3 - Let G(x,y) be “ x2y .” Indicate which of the...Ch. 3.3 - The following statement is true: “ nonzero number...Ch. 3.3 - The following statement is true: “ real number x,...Ch. 3.3 - Prob. 5ESCh. 3.3 - The statements in exercise 5-8 refer to the Tarski...Ch. 3.3 - Prob. 7ESCh. 3.3 - This statements is exercised 5-8 refer to the...Ch. 3.3 - Prob. 9ESCh. 3.3 - This exercise refers to Example 3.3.3. Determine...Ch. 3.3 - Let Sbe the set of students at your school, let M...Ch. 3.3 - Let D = E ={-2,-1,0,1,2}. Write negations for each...Ch. 3.3 - Prob. 13ESCh. 3.3 - Prob. 14ESCh. 3.3 - Prob. 15ESCh. 3.3 - Prob. 16ESCh. 3.3 - Prob. 17ESCh. 3.3 - Prob. 18ESCh. 3.3 - Prob. 19ESCh. 3.3 - Recall that reversing that order of the...Ch. 3.3 - For each of following equators, determinewhich of...Ch. 3.3 - Prob. 22ESCh. 3.3 - In 22 and 23, rewrite each statement without using...Ch. 3.3 - Prob. 24ESCh. 3.3 - Each statement in 25—28 refers to Tarski world of...Ch. 3.3 - Each statement in 25—28 refers to Tarski world of...Ch. 3.3 - Prob. 27ESCh. 3.3 - Prob. 28ESCh. 3.3 - Prob. 29ESCh. 3.3 - Prob. 30ESCh. 3.3 - Prob. 31ESCh. 3.3 - Prob. 32ESCh. 3.3 - Prob. 33ESCh. 3.3 - In 33-39(a) rewrite the statement formally using...Ch. 3.3 - In 33-39(a) rewrite the statement formally using...Ch. 3.3 - In 33-39(a) rewrite the statement formally using...Ch. 3.3 - In 33-39(a) rewrite the statement formally using...Ch. 3.3 - In 33-39(a) rewrite the statement formally using...Ch. 3.3 - Prob. 39ESCh. 3.3 - Prob. 40ESCh. 3.3 - Indicate which of the following statements are...Ch. 3.3 - Write the negation of the definition of limit of a...Ch. 3.3 - The following is the definition for limxaf(x)=L ....Ch. 3.3 - The notation ! stands for the words “There exists...Ch. 3.3 - Suppose that P(x) is a predicate and D is the...Ch. 3.3 - In 46—54, refer to the Tarski world given in...Ch. 3.3 - Prob. 47ESCh. 3.3 - Prob. 48ESCh. 3.3 - In 46—54, refer to the Tarski world given in...Ch. 3.3 - In 46—54, refer to the Tarski world given in...Ch. 3.3 - Y13In 46—54, refer to the Tarski world given in...Ch. 3.3 - Prob. 52ESCh. 3.3 - Prob. 53ESCh. 3.3 - Prob. 54ESCh. 3.3 - Let P(x)and Q(x) be predicates and suppose D is...Ch. 3.3 - Let P(x) and Q(x) be predicates and suppose D is...Ch. 3.3 - Let P(x) and Q(x) be predicates and suppose D is...Ch. 3.3 - Let P(x) and Q(x) be predicates and suppose D is...Ch. 3.3 - Prob. 59ESCh. 3.3 - In 59-61, find the answers Prolog would give if...Ch. 3.3 - Prob. 61ESCh. 3.4 - The rule of universal instantiation says that if...Ch. 3.4 - If the first two premises of universal modus...Ch. 3.4 - Prob. 3TYCh. 3.4 - If the first two premised of universal...Ch. 3.4 - Prob. 5TYCh. 3.4 - Prob. 1ESCh. 3.4 - Prob. 2ESCh. 3.4 - Prob. 3ESCh. 3.4 - real numbers r, a, and b, if b, if r is positive,...Ch. 3.4 - Prob. 5ESCh. 3.4 - Prob. 6ESCh. 3.4 - Some of the arguments in 7-18 are valid by...Ch. 3.4 - Prob. 8ESCh. 3.4 - Prob. 9ESCh. 3.4 - Prob. 10ESCh. 3.4 - Some of the arguments in 7—18 are valid by...Ch. 3.4 - Some of the arguments in 7—18 are valid by...Ch. 3.4 - Some of the arguments in 7-18 are valid by...Ch. 3.4 - Some of the arguments in 7-18 are valid by...Ch. 3.4 - Some of the arguments in 7-18 are valid by...Ch. 3.4 - Some of the arguments in 7-18 are valid by...Ch. 3.4 - Prob. 17ESCh. 3.4 - Some of the arguments in 7-18 are valid by...Ch. 3.4 - Rewrite the statement “No good cars are cheap” in...Ch. 3.4 - Use a diagram to shoe that the following argument...Ch. 3.4 - Indicate whether the arguments in 21-27 are valid...Ch. 3.4 - Indicate whether the arguments in 21-27 are valid...Ch. 3.4 - Prob. 23ESCh. 3.4 - Indicate whether the arguments in 21-27 are valid...Ch. 3.4 - Prob. 25ESCh. 3.4 - Prob. 26ESCh. 3.4 - Prob. 27ESCh. 3.4 - In exercises 28-32, reorder the premises in each...Ch. 3.4 - In exercises 28-32, reorder the premises in each...Ch. 3.4 - In exercises 28-32, reorder the premises in each...Ch. 3.4 - Prob. 31ESCh. 3.4 - In exercises 28-32, reorder the premises in each...Ch. 3.4 - Prob. 33ESCh. 3.4 - In 33 and 34 a single conclusion follows when all...Ch. 3.4 - Prob. 35ESCh. 3.4 - Derives the validity of universal form of part(a)...
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