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 3.4, Problem 7E
a.Give an example of sets A and B of real numbers such that
b.For sets A and B such that
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7. Show that for R sufficiently large, the polynomial P(z) in Example 3, Sec. 5, satisfies
the inequality
|P(z)| R.
Suggestion: Observe that there is a positive number R such that the modulus of
each quotient in inequality (9), Sec. 5, is less than |an|/n when |z| > R.
9. Establish the identity
1-
1+z+z² +
2n+1
...
+z" =
1- z
(z1)
and then use it to derive Lagrange's trigonometric identity:
1
1+ cos cos 20 +... + cos no =
+
2
sin[(2n+1)0/2]
2 sin(0/2)
(0 < 0 < 2л).
Suggestion: As for the first identity, write S = 1+z+z² +...+z" and consider
the difference S - zS. To derive the second identity, write z =
eie in the first one.
8. Prove that two nonzero complex numbers z₁ and Z2 have the same moduli if and only if
there are complex numbers c₁ and c₂ such that Z₁ = c₁C2 and Z2 = c1c2.
Suggestion: Note that
(i≤
exp (101+0) exp (01-02)
and [see Exercise 2(b)]
2
02
Ꮎ
-
= = exp(i01)
exp(101+0) exp (i 01 - 02 ) = exp(102).
i
2
2
Chapter 3 Solutions
A Transition to Advanced Mathematics
Ch. 3.1 - Let 3 and 6 be the sets of integer multiples of 3...Ch. 3.1 - Let (3,+) and (6,+) be the groups in Exercise 10,...Ch. 3.1 - Let ({a,b,c},o) be the group with the operation...Ch. 3.1 - (a)Prove that the function f:1824 given by f(x)=4x...Ch. 3.1 - Define f:1512 by f(x)=4x. Prove that f is a...Ch. 3.1 - Let (G,) and (H,*) be groups, i be the identity...Ch. 3.1 - Prob. 7ECh. 3.1 - Prob. 8ECh. 3.1 - Prove that the relation of isomorphism is an...Ch. 3.1 - Prob. 10E
Ch. 3.1 - Prove that if G is a group and H is a subgroup of...Ch. 3.1 - Prob. 12ECh. 3.1 - Prob. 13ECh. 3.1 - Prob. 14ECh. 3.1 - Prob. 15ECh. 3.1 - Prob. 16ECh. 3.1 - Prob. 17ECh. 3.2 - (a)Show that any two groups of order 2 are...Ch. 3.2 - (a)Show that the function h: defined by h(x)=3x is...Ch. 3.2 - Let R be the equivalence relation on ({0}) given...Ch. 3.2 - Let (R,+,) be an integral domain. Prove that 0 has...Ch. 3.2 - Complete the proof of Theorem 6.5.5. That is,...Ch. 3.2 - Prob. 6ECh. 3.2 - Assign a grade of A (correct), C (partially...Ch. 3.2 - Prob. 8ECh. 3.2 - Prob. 9ECh. 3.2 - Use the method of proof of Cayley's Theorem to...Ch. 3.2 - Prob. 11ECh. 3.2 - Assign a grade of A (correct), C (partially...Ch. 3.2 - Prob. 13ECh. 3.2 - Define on by setting (a,b)(c,d)=(acbd,ad+bc)....Ch. 3.2 - Prob. 15ECh. 3.2 - Let f:(A,)(B,*) and g:(B,*)(C,X) be OP maps. Prove...Ch. 3.2 - Prob. 17ECh. 3.2 - Let Conj: be the conjugate mapping for complex...Ch. 3.2 - Prove the remaining parts of Theorem 6.4.1.Ch. 3.3 - Let 3={3k:k}. Apply the Subring Test (Exercise...Ch. 3.3 - Use these exercises to check your understanding....Ch. 3.3 - Use these exercises to check your understanding....Ch. 3.3 - Use these exercises to check your understanding....Ch. 3.3 - Use these exercises to check your understanding....Ch. 3.3 - Prob. 6ECh. 3.3 - Use the definition of “divides” to explain (a) why...Ch. 3.3 - Prob. 8ECh. 3.3 - Prob. 9ECh. 3.3 - Complete the proof that for every m,(m+,) is a...Ch. 3.3 - Define addition and multiplication on the set ...Ch. 3.3 - Prob. 12ECh. 3.3 - Let (R,+,) be a ring and a,b,R. Prove that b+(a)...Ch. 3.3 - Prove the remaining parts of Theorem 6.5.3: For...Ch. 3.3 - We define a subring of a ring in the same way we...Ch. 3.4 - Prob. 1ECh. 3.4 - Prob. 2ECh. 3.4 - If possible, give an example of a set A such that...Ch. 3.4 - Let A. Prove that if sup(A) exists, then...Ch. 3.4 - Let A and B be subsets of . Prove that if sup(A)...Ch. 3.4 - a.Give an example of sets A and B of real numbers...Ch. 3.4 - a.Give an example of sets A and B of real numbers...Ch. 3.4 - An alternate version of the Archimedean Principle...Ch. 3.4 - Prob. 9ECh. 3.4 - Prob. 10ECh. 3.4 - Prob. 11ECh. 3.4 - Prob. 12ECh. 3.5 - Prob. 1ECh. 3.5 - Prob. 2ECh. 3.5 - Let A be a subset of . Prove that the set of all...Ch. 3.5 - Prob. 4ECh. 3.5 - Let be an associative operation on nonempty set A...Ch. 3.5 - Suppose that (A,*) is an algebraic system and * is...Ch. 3.5 - Let (A,o) be an algebra structure. An element lA...Ch. 3.5 - Let G be a group. Prove that if a2=e for all aG,...Ch. 3.5 - Give an example of an algebraic structure of order...Ch. 3.5 - Prove that an ordered field F is complete iff...Ch. 3.5 - Prove that every irrational number is "missing"...Ch. 3.5 - Find two upper bounds (if any exits) for each of...Ch. 3.5 - Prob. 13ECh. 3.5 - Prob. 14ECh. 3.5 - Prob. 15ECh. 3.5 - Let A and B be subsets of . Prove that if A is...Ch. 3.5 - Prob. 17ECh. 3.5 - Prob. 18ECh. 3.5 - Give an example of a set A for which both A and Ac...Ch. 3.5 - Prob. 20ECh. 3.5 - Prob. 21ECh. 3.5 - Prob. 22E
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- 1) Compute the inverse of the following matrix. 0 1 1 A = 5 1 -1 2-3 -3arrow_forward2) Consider the matrix M = [1 2 3 4 5 0 2 3 4 5 00345 0 0 0 4 5 0 0 0 0 5 Determine whether the following statements are True or False. A) M is invertible. B) If R5 and Mx = x, then x = 0. C) The last row of M² is [0 0 0 0 25]. D) M can be transformed into the 5 × 5 identity matrix by a sequence of elementary row operations. E) det (M) 120 =arrow_forward3) Find an equation of the plane containing (0,0,0) and perpendicular to the line of intersection of the planes x + y + z = 3 and x y + z = 5. -arrow_forward
- 1) In the xy-plane, what type of conic section is given by the equation - √√√(x − 1)² + (y − 1)² + √√√(x + 1)² + (y + 1)² : - = 3?arrow_forward3) Let V be the vector space of all functions f: RR. Prove that each W below is a subspace of V. A) W={f|f(1) = 0} B) W = {f|f(1) = ƒ(3)} C) W={ff(x) = − f(x)}arrow_forwardTranslate the angument into symbole from Then determine whether the argument is valid or Invalid. You may use a truth table of, it applicable compare the argument’s symbolic form to a standard valid or invalid form. pot out of bed. The morning I did not get out of bed This moring Mat woke up. (1) Cidt the icon to view tables of standard vald and braild forms of arguments. Let prepresent."The morning Must woke up "and let a represent “This morning I got out of bed.” Seled the cared choice below and II in the answer ber with the symbolic form of the argument (Type the terms of your expression in the same order as they appear in the original expression) A. The argument is valid In symbolic form the argument is $\square $ B. The angunent is braid In symbolic form the argument is $\square $arrow_forward
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