Discrete Mathematics with Graph Theory (Classic Version) (3rd Edition) (Pearson Modern Classics for Advanced Mathematics Series)
3rd Edition
ISBN: 9780134689555
Author: Edgar Goodaire, Michael Parmenter
Publisher: PEARSON
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Chapter 1.2, Problem 5TFQ
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Find parametric equations for the line through the point (0, 2, 2) that is perpendicular to the line x=3+t, y 2-t, z=3t and intersects this line. (Use the parameter t.)
(x(t), y(t), z(t)) = ( t,2 — t,2 + 2t
)
Let P be a point not on the line L that passes through the points Q and R. The distance d from the point P to the line L is
|a x bl
d
|a|
where a =
QR and b = QP.
Use the above formula to find the distance from the point to the given line.
d =
(0, 1, 3); x = 2t, y = 6 - 2t, z = 3 + t
Let L₁ be the line through the origin and the point (2, 0, -1). Let L₂ be the line through the points (1, -1, 1) and (6, 1, 5). Find the distance between L1 and L2.
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Chapter 1 Solutions
Discrete Mathematics with Graph Theory (Classic Version) (3rd Edition) (Pearson Modern Classics for Advanced Mathematics Series)
Ch. 1.1 - True/False Questions
“” means “”
Ch. 1.1 - A truth table based on four simple statements...Ch. 1.1 - True/False Questions
2. If is true, then is also...Ch. 1.1 - If p and q are both false, the truth value of...Ch. 1.1 - If pq is false, the truth value of (pq)(pq) is...Ch. 1.1 - pq andqp are logically equivalent.Ch. 1.1 - True/False Questions
7. A statement and its...Ch. 1.1 - (pq)(pq) is a tautology.Ch. 1.1 - True/False Questions
9. If B is a tautology and A...Ch. 1.1 - True/False Questions
10. If A and B are both...
Ch. 1.1 - Construct a truth table for each of the following...Ch. 1.1 - (a) If pq is false, determine the truth value of...Ch. 1.1 - 3. Determine the truth value for
when are all...Ch. 1.1 - 4. Repeat Exercise 3 in the case where are all...Ch. 1.1 - 5. (a) Show that is a tautology.
(b) Show that ...Ch. 1.1 - Prob. 6ECh. 1.1 - Prob. 7ECh. 1.1 - Prob. 8ECh. 1.1 - Prob. 9ECh. 1.1 - 10. (a) Show that the statement is not logically...Ch. 1.1 - 11. If and are statements, then the compound...Ch. 1.2 - True/False Questions
Two statements A and B are...Ch. 1.2 - True/False Questions
2. “A B” and “A B” mean the...Ch. 1.2 - True/False Questions
3. () () for any statement ....Ch. 1.2 - True/False Questions
4. for any statements
Ch. 1.2 - (p(qr))((pq)(pr)) for any statements p,q,r.Ch. 1.2 - ((pq))((p)(q)) for any statements p,q.Ch. 1.2 - If A Band C is any statement, then (A C) (B ...Ch. 1.2 - True/False Questions
8. is in disjunctive normal...Ch. 1.2 - (pq(r))((p)(q)(r)) is in disjunctive normal form.Ch. 1.2 - True/False Questions
10. Disjunctive normal form...Ch. 1.2 - Prob. 1ECh. 1.2 - (a) Show that p[(pq)] is a tautology. (b) What is...Ch. 1.2 - Simplify each of the following statements. (a)...Ch. 1.2 - 4. Using truth tables, verify the following...Ch. 1.2 - 5. Using the properties in the text together with...Ch. 1.2 - Prove that the statements (p(q))q and (p(q))p are...Ch. 1.2 - Prob. 7ECh. 1.2 - Prob. 8ECh. 1.2 - Prob. 9ECh. 1.2 - Express each of the following statements in...Ch. 1.2 - Find out what you can about Augustus De Morgan and...Ch. 1.3 - True/False Questions
An argument is valid if,...Ch. 1.3 - Prob. 2TFQCh. 1.3 - Prob. 3TFQCh. 1.3 - True/False Questions
4. De Morgan’s laws are two...Ch. 1.3 - The chain rule has pq and qr as its premises.Ch. 1.3 - Prob. 6TFQCh. 1.3 - Prob. 7TFQCh. 1.3 - Prob. 8TFQCh. 1.3 - Prob. 9TFQCh. 1.3 - Prob. 10TFQCh. 1.3 - Determine whether or not each of the following...Ch. 1.3 - 2. Verify that each of the five rules of inference...Ch. 1.3 - Verify that each of the following arguments is...Ch. 1.3 - Test the validity of each of the following...Ch. 1.3 - 5. Determine the validity of each of the following...Ch. 1.3 - Prob. 6ECh. 1.3 - Prob. 7ECh. 1.3 - Prob. 8ECh. 1.3 - Prob. 9ECh. 1.3 - 10. What language is being used when we say “modus...Ch. 1 - Construct a truth table for the compound statement...Ch. 1 - Determine the truth value of [p(q((r)s))](rt),...Ch. 1 - 3. Determine whether each statement is a...Ch. 1 - Two compound statements A and B have the property...Ch. 1 - 5. (a) Suppose A, B, and C are compound statements...Ch. 1 - Establish the logical equivalence of each of the...Ch. 1 - 7. Express each of the following statements in...Ch. 1 - Determine whether each of the following arguments...Ch. 1 - Discuss the validity of the argument pq(p)r Purple...Ch. 1 - 10. Determine the validity of each of the...
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Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.Similar questions
- (a) Let P be a point not on the line L that passes through the points Q and R. Show that the distance d from the point P to the line L is |a x bl |a| d where a = QR and b = QP. This answer has not been graded yet. (b) Use the formula in part (a) to find the distance from the point P(1, 1, 1) to the line through Q(0, 7, 6) and R(-1, 2, 6). 29.65arrow_forwardFind the area of the parallelogram with vertices K(1, 2, 2), L(1, 5, 4), M(6, 10, 4), and N(6, 7, 2).arrow_forwardFind the area of the parallelogram with vertices A(-5, 4), B(-3, 7), C(1, 5), and D(-1, 2).arrow_forward
- Find an equation of the plane. The plane through the point (8, 0, 4) and perpendicular to the line x = 3t, y = 6-t, z = 7 + 4tarrow_forwardFind an equation of the plane. The plane that passes through the line of intersection of the planes x-z=3 and y + 4z1 and is perpendicular to the plane x + y 2 = 4 5x+4y+3z 27 - Need Help? Read Itarrow_forwardFind the volume of the parallelepiped with adjacent edges PQ, PR, PS. P(3, 0, -2), Q(6, 2, 0), R(6, -1, 1), S(3, -3, 1) cubic units Need Help? Read It Watch Itarrow_forward
- Find a vector equation and parametric equations for the line. (Use the parameter t.) The line through the point (8, -5, 2) and parallel to the vector 2 - 3 r(t) = (x(t), y(t), z(t)) = (3,2.7,3.1)+(33, − 1) -arrow_forwardFind the cross product a x b. a = (t, t², t³), b = (1, 4t, 9t²)arrow_forwardFind the distance from the point to the given plane. (-3, 3, 2), x-2y-4z = 8arrow_forward
- Find the volume of the parallelepiped determined by the vectors a, b, and c. a = (6,2,-3), с b = (0, 3, 3), c = (6, -2, 4) cubic units Need Help? Read It Watch Itarrow_forwardFind the cosine of the angle between the planes x + y + z = 0 and x + 2y + 3z = 9.arrow_forwardFind a vector equation for the line segment from (3, -3, 4) to (7, 3, 1). (Use the parameter t.) r(t) = Need Help? Read It Watch Itarrow_forward
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