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 4.4, Problem 8E
Give an example of
- a bounded sequence that is not convergent.
- an increasing sequence that is not convergent.
- a convergent sequence that is not monotone.
- a divergent sequence x such that the sequence whose nth term is
- an increasing sequence bounded above by 2 that does not converge to 2.
- an increasing sequence that converges to
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7. [10 marks]
Let G
=
(V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a
cycle in G on which x, y, and z all lie.
(a) First prove that there are two internally disjoint xy-paths Po and P₁.
(b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which
x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that
there are three paths Qo, Q1, and Q2 such that:
⚫each Qi starts at z;
• each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are
distinct;
the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex
2) and are disjoint from the paths Po and P₁ (except at the end vertices wo,
W1, and w₂).
(c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and
z all lie. (To do this, notice that two of the w; must be on the same Pj.)
Chapter 4 Solutions
A Transition to Advanced Mathematics
Ch. 4.1 - Find two upper bounds (if any exits) for each of...Ch. 4.1 - Assign a grade of A (correct), C (partially...Ch. 4.1 - Prob. 3ECh. 4.1 - Prob. 4ECh. 4.1 - Let A and B be subsets of . Prove that if A is...Ch. 4.1 - Let x be an upper bound for A. Prove that if xy,...Ch. 4.1 - Let A. Prove that if A is bounded above, then Ac...Ch. 4.1 - Give an example of a set A for which both A and Ac...Ch. 4.1 - Let A. Prove that if sup(A) exists, then it is...Ch. 4.1 - Formulate and prove a characterization of greatest...
Ch. 4.1 - If possible, give an example of a set A such that...Ch. 4.1 - Let A. Prove that if sup(A) exists, then...Ch. 4.1 - Let A and B be subsets of . Prove that if sup(A)...Ch. 4.1 - (a)Give an example of sets A and B of real numbers...Ch. 4.1 - (a)Give an example of sets A and B of real numbers...Ch. 4.1 - An alternate version of the Archimedean Principle...Ch. 4.1 - Prob. 17ECh. 4.1 - Prove that an ordered field F is complete iff...Ch. 4.1 - Prove that every irrational number is "missing"...Ch. 4.2 - Let A and B be compact subsets of . Use the...Ch. 4.2 - Prob. 2ECh. 4.2 - Prob. 3ECh. 4.2 - Prob. 4ECh. 4.2 - Assign a grade of A (correct), C (partially...Ch. 4.2 - For real numbers x,1,2,...n, describe i=1nN(x,i)....Ch. 4.2 - State the definition of continuity of the function...Ch. 4.2 - Find the set of interior point for each of these...Ch. 4.2 - Suppose that x is an interior point of a set A....Ch. 4.2 - Let AB. Prove that if sup(A) and sup(B) both...Ch. 4.2 - Let Abe a nonempty collection of closed subsets of...Ch. 4.2 - Prob. 12ECh. 4.2 - Prob. 13ECh. 4.2 - Prob. 14ECh. 4.2 - Prob. 15ECh. 4.2 - Prob. 16ECh. 4.2 - Prove Lemma 7.2.4.Ch. 4.2 - Which of the following subsets of are compact? ...Ch. 4.2 - Give an example of a bounded subset of and a...Ch. 4.3 - Let A and F be sets of real numbers, and let F be...Ch. 4.3 - In the proof of Theorem 7.3.1 that =, it is...Ch. 4.3 - Assign a grade of A (correct), C (partially...Ch. 4.3 - Prove that 7 is an accumulation point for [3,7). 5...Ch. 4.3 - Find an example of an infinite subset of that has...Ch. 4.3 - Find the derived set of each of the following...Ch. 4.3 - Let S=(0,1]. Find S(Sc).Ch. 4.3 - Prob. 8ECh. 4.3 - (a)Prove that if AB, then AB. (b)Is the converse...Ch. 4.3 - Show by example that the intersection of...Ch. 4.3 - Prob. 11ECh. 4.3 - Prob. 12ECh. 4.3 - Let a, b. Prove that every closed interval [a,b]...Ch. 4.3 - Prob. 14ECh. 4.3 - Prob. 15ECh. 4.4 - Prob. 1ECh. 4.4 - Prove that if x is an interior point of the set A,...Ch. 4.4 - Recall from Exercise 11 of Section 4.6 that the...Ch. 4.4 - A sequence x of real numbers is a Cauchy* sequence...Ch. 4.4 - Prob. 5ECh. 4.4 - Assign a grade of A (correct), C (partially...Ch. 4.4 - Prob. 7ECh. 4.4 - Give an example of a bounded sequence that is not...Ch. 4.4 - Prob. 9ECh. 4.4 - Let A and B be subsets of . Prove that (AB)=AB....Ch. 4.5 - For the sequence y defined in the proof of Theorem...Ch. 4.5 - Prob. 2ECh. 4.5 - Prob. 3ECh. 4.5 - Let I be a sequence of intervals. Then for each...Ch. 4.5 - Prob. 5ECh. 4.5 - Prob. 6ECh. 4.5 - Find all divisors of zero in 14. 15. 10. 101.Ch. 4.5 - Prob. 8ECh. 4.5 - Suppose m and m2. Prove that 1 and m1 are distinct...Ch. 4.5 - Prob. 10ECh. 4.5 - Prob. 11ECh. 4.5 - Determine whether each sequence is monotone. For...Ch. 4.5 - Prob. 13ECh. 4.5 - Complete the proof that xn=(1+1n)n is increasing...Ch. 4.5 - Prob. 15ECh. 4.5 - Prob. 16ECh. 4.5 - Prob. 17ECh. 4.6 - Prob. 1ECh. 4.6 - Repeat Exercise 2 with the operation * given by...Ch. 4.6 - Prob. 3ECh. 4.6 - Let m,n and M=A:A is an mn matrix with real number...Ch. 4.6 - Let be an associative operation on nonempty set A...Ch. 4.6 - Let be an associative operation on nonempty set A...Ch. 4.6 - Suppose that (A,*) is an algebraic system and * is...Ch. 4.6 - Let (A,o) be an algebra structure. An element lA...Ch. 4.6 - Let G be a group. Prove that if a2=e for all aG,...Ch. 4.6 - Prob. 10ECh. 4.6 - Complete the proof of Theorem 6.1.4. First, show...Ch. 4.6 - Prob. 12ECh. 4.6 - Prob. 13ECh. 4.7 - Give an example of an algebraic structure of order...Ch. 4.7 - Let G be a group. Prove that G is abelian if and...Ch. 4.7 - Prob. 3ECh. 4.7 - (a)In the group G of Exercise 2, find x such that...Ch. 4.7 - Show that (,), with operation # defined by...Ch. 4.7 - Let m be a prime natural number and a(Um,). Prove...Ch. 4.7 - Prob. 7ECh. 4.7 - Prob. 8ECh. 4.7 - Prob. 9E
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Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Similar questions
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