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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Textbook Question
Chapter 0.2, Problem 4E
Consider the following two statements:
A: The square of every real number is positive.
B: There exists a real number with a square less than
Each of these statements is false. One can easily be proved false with a counterexample. The other requires a (short) direct proof. Which is which? Explain your answer.
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Prove:
{a,{a,b}}={c,{c,d}} ⟺ a=c and b=d.
3. Define a function T : R³ → R² by
T(x, y, z) = (x + y + z, x + 2y − 3z).
(1) Show that T is a linear transformation.
—
(2) Find all vectors in the kernel of T.
(3) Show that T is onto.
(4) Find the matrix representation of T relative to the standard ba-
sis of R³ and R².
5. Use the Gram-Schmidt process to find an orthonormal basis for the
subspace of R4 spanned by the vectors
(1,0,1,1),
(1,0,1,0),
(0, 0, 1, 1).
Chapter 0 Solutions
Discrete Mathematics with Graph Theory (Classic Version) (3rd Edition) (Pearson Modern Classics for Advanced Mathematics Series)
Ch. 0.1 - pandq is false if pandq are false.Ch. 0.1 - If pandq is false, then both pandq are false.Ch. 0.1 - True/False Questions
3. It is possible for both “”...Ch. 0.1 - True/False Questions
4. It is possible for both “”...Ch. 0.1 - The implication If 22=5, then 32=9 is true.Ch. 0.1 - The negation of a=b=0 is ab0.Ch. 0.1 - The converse of the implication in Question 5 is...Ch. 0.1 - True/False Questions
8. The double implication “...Ch. 0.1 - It is possible for both an implication and its...Ch. 0.1 - The statement Some frogs have red toes makes use...
Ch. 0.1 - The negation of an existential quantifier is its...Ch. 0.1 -
Classify each of the following statements as...Ch. 0.1 -
Classify each of the following statements as...Ch. 0.1 - 3. Rewrite each of the following statements so...Ch. 0.1 -
4. Determine whether each of the following...Ch. 0.1 - Write down the negation of each of the following...Ch. 0.1 -
6. Write down the converse and contrapositive of...Ch. 0.1 - Rewrite each of the following statements using the...Ch. 0.1 - Is it possible for both an implication and its...Ch. 0.1 - On page 4 of the text, we stated as more or less...Ch. 0.2 - If you want to prove a statement is true, it is...Ch. 0.2 - True/False Questions
2. If you want to prove a...Ch. 0.2 - The sentence A is a sufficient condition for Bis...Ch. 0.2 - True/False Questions
4. If A B, BC, CD, and CA...Ch. 0.2 - True/False Questions
5. If A B, BC, CD, and CA...Ch. 0.2 - The contrapositive of A Bis B A.Ch. 0.2 - A Bis true if and only if its contrapositive is...Ch. 0.2 - True/False Questions
8. is a rational number.
Ch. 0.2 - True/False Questions
9. 3.141 is a rational...Ch. 0.2 - True/False Questions
10. If and are irrational...Ch. 0.2 - True/False Questions
11. The statement “Every...Ch. 0.2 - The statement There exists an irrational number...Ch. 0.2 - What is the hypothesis and what is the conclusion...Ch. 0.2 - 2. In each part of Exercise 1, what condition is...Ch. 0.2 - Exhibit a counterexample to each of the following...Ch. 0.2 - Consider the following two statements: A: The...Ch. 0.2 - Determine whether the following implication is...Ch. 0.2 - State the converse of the implication in Exercise...Ch. 0.2 - 7. Answer Exercise 5 with replaced by .
Ch. 0.2 - Consider the statement A: If n is an integer, nn+1...Ch. 0.2 - 9. Let be an integer greater than 1 and consider...Ch. 0.2 - 10. A theorem in calculus states that every...Ch. 0.2 - 11. Let be an integer, . A certain mathematical...Ch. 0.2 - Consider the assertions A: For every real number...Ch. 0.2 - Answer Exercise 12 with A and B as follows. A:...Ch. 0.2 - 14. Answer true or false and supply a direct proof...Ch. 0.2 - Prove that n an even integer n2+3n is an even...Ch. 0.2 - 16. (a) Let be an integer. Show that either or...Ch. 0.2 - 17. Provide a direct proof that is odd for all...Ch. 0.2 - Prove that 2x24x+30 for any real number x.Ch. 0.2 - 19. Let and be integers. By examining the four...Ch. 0.2 - Let n be an integer. Prove that n2 is even if and...Ch. 0.2 - Prob. 21ECh. 0.2 - Prove that if n is an odd integer then there is an...Ch. 0.2 - 23. Prove that if is an odd integer, there is an...Ch. 0.2 - 24. Prove that there exists no smallest positive...Ch. 0.2 - 25. Let be the product of positive integers and ....Ch. 0.2 - 26. (For students who have studied linear algebra)...Ch. 0.2 - 27. (a) Suppose and are integers such that . Prove...Ch. 0.2 - Suppose a and b are integers such that a+b+ab=0....Ch. 0.2 - Suppose a is an irrational number. Prove that 1a...Ch. 0.2 - 30. Suppose that is a rational number and that is...Ch. 0.2 - Prob. 31ECh. 0.2 - 32. Find a proof or exhibit a counterexample to...Ch. 0.2 - Prob. 33ECh. 0.2 - Prob. 34ECh. 0.2 - Prob. 35ECh. 0.2 - Prob. 36ECh. 0.2 - Prob. 37ECh. 0.2 - Prove that there exist irrational numbers a and b...Ch. 0 - State, with a reason, whether each of the...Ch. 0 - Prob. 2RECh. 0 - 3. Write down the converse, the contrapositive and...Ch. 0 - Prob. 4RECh. 0 - Prob. 5RECh. 0 - Prob. 6RECh. 0 - Prob. 7RECh. 0 - Prob. 8RECh. 0 - 9. Let be an integer. Prove that is odd if and...Ch. 0 - Give a direct proof of the fact that a25a+6 is...Ch. 0 - Prob. 11RECh. 0 - Prob. 12RECh. 0 - 13. Prove, by way of contradiction, that if is a...Ch. 0 - Prob. 14RECh. 0 - Prob. 15RECh. 0 - Prob. 16RECh. 0 - Prob. 17RECh. 0 - Prob. 18RECh. 0 - Each of the integers 31, 331, 3331, 33331, 333331,...
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- 4. Show that B = {(1, 1, 1), (1, 1, 0), (0, 1, 1)} is a basis for R³. Find the coordinate vector of (1, 2, 3) relative to the basis B.arrow_forward1. Solve the following system of linear equations. x12x2 3x3 + 4x4 +5x5 x1+2×2+4x3 + 3x4 +7x5 x12x2 + 2x3 + 5x4 + 4x5 Write your solution in parametric form. = 6 = 5 = 9.arrow_forward6. Consider the matrix B = 3 2 -3 -3-49 -1-2 5 Find an invertible matrix P and a diagonal matrix D such that B = PDP-¹.arrow_forward
- 2. Consider the matrix 1 3 3 3 A: = 1 4 3 1 3 4 Find the determinant |A| and the inverse matrix A-1.arrow_forwardLet tpQ be a translation through the vector PQ and RA be a rotation centered at A through angle 0. What can you say about the products (tp) (RA,) and (R₁₁)(t)? Is it possible to simplify either of these products at all? Please explain thoroughly.arrow_forwardLet Râ‚ and RB, be rotations and consider the product RA, RB,. What do you believe must Ꭺ,0 Φ be true about A, B,0, and & if RÃRÂ, is a translation? What do you believe must be true about A, B,0, and if RÃR‚ is a rotation? Please explain thoroughly. Ꭺ,0 B,arrow_forward
- Let m be the line given by the equation y = 0. Let n be the line given by the equation y = =2x, rprnrm? If = and let p be the line given by the equation x = 0. Is there a line զ such that ra so, please find the equation of q and describe any relationships you notice between line q and the other three lines. If not, please explain why not. Please describe your process thoroughly.arrow_forwardConsider the lines through the origin x, m, n, l, and y, where x represents the x-axis, y represents the y-axis, m has angle 0 from the positive x-axis, n has angle & from the positive x-axis, and has angle y from the positive x-axis, with 0 < 0 < & < y < 90°. It is possible to write the product ryrernrmrx as a single reflection rk. Determine the angle between the line k and the positive x-axis in terms of angles 0, 0, and y.arrow_forwardCompute analytically the equation of the line m' obtained from rotating the line m: 5x+2y about the point (-3, -1). Please thoroughly describe your process. = 3arrow_forward
- Consider the glide reflection rk™n™m given by the lines m : −4x + 3y = 9, n: −4x + 3y = −16, and k 3x+4y = : 12. Determine an analytic representation of this isometry. That is, given any point (x, y) = R², derive a formula for the image (x', y') of this point under the glide reflection. Explain your thought process for deriving this representation.arrow_forwardShow analytically that if m is a line through the origin with inclination angle from the positive x-axis and if n is a line through the origin with inclination angle 0 from the positive x-axis, is a rotation about the origin by angle 2(0 – 6). then rnrmarrow_forwardLet A, B E R² and let 0 = 180°. Prove that RB, RA, is a translation through vector 2AB.arrow_forward
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