Explanation For any number a 1 , a 2 ∈ a , the function f ( a ) is one to one if f ( a 1 ) = f ( a 2 ) ⇒ a 1 = a 2 . For any number a 1 , a 2 ∈ ℝ , the function f ( a ) is increasing if a 1 < a 2 ⇒ f ( a 1 ) < f ( a 2 ) and decreasing if a 1 < a 2 ⇒ f ( a 1 ) > f ( a 2 ) . Case I: Increasing Let a 1 , a 2 ∈ ℝ such that a 1 ≠ a 2 ⇒ a 1 < a 2 or a 1 > a 2 ⇒ f ( a 1 ) < f ( a 2 ) or f ( a 1 ) >

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied unde…

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Chapter 4.3, Problem 11E

To determine

To prove: that if a real valued function is increasing or decreasing on ℝ then the function is one to one.

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Chapter 4.3, Problem 10E

Chapter 4.3, Problem 12E

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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. 3E Ch. 4.1 - Prob. 4E Ch. 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. 17E Ch. 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. 2E Ch. 4.2 - Prob. 3E Ch. 4.2 - Prob. 4E Ch. 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. 12E Ch. 4.2 - Prob. 13E Ch. 4.2 - Prob. 14E Ch. 4.2 - Prob. 15E Ch. 4.2 - Prob. 16E Ch. 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. 8E Ch. 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. 11E Ch. 4.3 - Prob. 12E Ch. 4.3 - Let a, b. Prove that every closed interval [a,b]...Ch. 4.3 - Prob. 14E Ch. 4.3 - Prob. 15E Ch. 4.4 - Prob. 1E Ch. 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. 5E Ch. 4.4 - Assign a grade of A (correct), C (partially...Ch. 4.4 - Prob. 7E Ch. 4.4 - Give an example of a bounded sequence that is not...Ch. 4.4 - Prob. 9E Ch. 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. 2E Ch. 4.5 - Prob. 3E Ch. 4.5 - Let I be a sequence of intervals. Then for each...Ch. 4.5 - Prob. 5E Ch. 4.5 - Prob. 6E Ch. 4.5 - Find all divisors of zero in 14. 15. 10. 101.Ch. 4.5 - Prob. 8E Ch. 4.5 - Suppose m and m2. Prove that 1 and m1 are distinct...Ch. 4.5 - Prob. 10E Ch. 4.5 - Prob. 11E Ch. 4.5 - Determine whether each sequence is monotone. For...Ch. 4.5 - Prob. 13E Ch. 4.5 - Complete the proof that xn=(1+1n)n is increasing...Ch. 4.5 - Prob. 15E Ch. 4.5 - Prob. 16E Ch. 4.5 - Prob. 17E Ch. 4.6 - Prob. 1E Ch. 4.6 - Repeat Exercise 2 with the operation * given by...Ch. 4.6 - Prob. 3E Ch. 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. 10E Ch. 4.6 - Complete the proof of Theorem 6.1.4. First, show...Ch. 4.6 - Prob. 12E Ch. 4.6 - Prob. 13E Ch. 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. 3E Ch. 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. 7E Ch. 4.7 - Prob. 8E Ch. 4.7 - Prob. 9E

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