4th Edition

ISBN: 9781119760061

Author: Burger

Publisher: WILEY

Concept explainers

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

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Textbook Question

Chapter 3.4, Problem 21MS

Adding another. Suppose that you have any infinite set (it could be really huge), and you wish to add one new element to the set. Prove that this new set (the old set with the new element thrown in) has the same cardinality as the original set. (Hint: The infinite set has a subset that is in one-to-one correspondence with the natural numbers. Use previous ideas to add the new element to this subset, and let the rest of the set correspond with itself.)

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Chapter 3.4, Problem 20MS

Chapter 3.4, Problem 22MS

Chapter 3 Solutions

HEART OF MATHEMATICS

Ch. 3.1 - Still the one. What is a one-to-one...Ch. 3.1 - Prob. 2MS Ch. 3.1 - Numerical nephwe. At a family gathering, your...Ch. 3.1 - Pile of packs. You walk into class late and notice...Ch. 3.1 - Bunch of balls. Your first job every morning at...Ch. 3.1 - The same, but unsure how much (H). We have used a...Ch. 3.1 - Taking stock (S). It turns out that there is a...Ch. 3.1 - Prob. 8MS Ch. 3.1 - Heres looking @ Â®. The following collections...Ch. 3.1 - Enough underwear. When Deb packs for a trip, she...

Ch. 3.1 - 791ZWV. Suppose a stranger tells you that the...Ch. 3.1 - 2452345. Suppose a stranger tells you that her...Ch. 3.1 - Social security (H). Is there a one-to-one...Ch. 3.1 - Testing one two three. A professor wishes to...Ch. 3.1 - Laundry day (ExH). Suppose you are given a bag of...Ch. 3.1 - Hair counts. Do there exist two nonbald people on...Ch. 3.1 - Social number (S). Social Security numbers contain...Ch. 3.1 - Prob. 18MS Ch. 3.1 - Dining hall blues. One day in Ralph P. Uke Dining...Ch. 3.1 - Dorm life(H). Every student at a certain college...Ch. 3.1 - Pigeonhole principle. Recall the Pigeonhole...Ch. 3.1 - Mother and child. Every child has one and only one...Ch. 3.1 - Coast to coast. Jessica is working part-time from...Ch. 3.1 - An interesting correspondence. Suppose you invest...Ch. 3.1 - Chicken Little. With increased attention to eating...Ch. 3.1 - Table for four. The table below shows a one-to-one...Ch. 3.1 - Square table. The table below shows a one-to-one...Ch. 3.2 - Au natural. Describe the set of natural numbers.Ch. 3.2 - Prob. 2MS Ch. 3.2 - Set setup. We can denote the natural numbers...Ch. 3.2 - Little or large. Which of the sets in Mindscape 3...Ch. 3.2 - A word you can count on. Define the cardinality of...Ch. 3.2 - Prob. 6MS Ch. 3.2 - Naturally even. Let E stand for the set of all...Ch. 3.2 - Fives take over. Let EIF be the set of all natural...Ch. 3.2 - Six times as much (EH). If we let N stand for the...Ch. 3.2 - Any times as much. If we let N stand for the set...Ch. 3.2 - Missing 3 (H). Let TIM be the set of all natural...Ch. 3.2 - One weird set. Let OWS (you figure it out) be the...Ch. 3.2 - Squaring off. Let S stand for the set of all...Ch. 3.2 - Counting Cubes (formerly Crows). Let C stand for...Ch. 3.2 - Reciprocals. Suppose R is the set defined by R={...Ch. 3.2 - Hotel Cardinality (formerly California) (H). It is...Ch. 3.2 - Hotel Cardinality continued. Given the scenario in...Ch. 3.2 - More Hotel C (EH). Given the scenario in Mindscape...Ch. 3.2 - So much sand. Prove that there cannot be an...Ch. 3.2 - Prob. 20MS Ch. 3.2 - Pruning sets. Suppose you have a set. If you...Ch. 3.2 - A natural prune. Describe a collection of numbers...Ch. 3.2 - Prune growth. Is it possible to remove things from...Ch. 3.2 - Same cardinality? Suppose we have two sets and we...Ch. 3.2 - Still the same? (S). Suppose we have two sets, and...Ch. 3.2 - Modest rationals (H). Devise and then describe a...Ch. 3.2 - A window of rationals. Using your answer to...Ch. 3.2 - Bowling ball barrel. Suppose you have infinitely...Ch. 3.2 - Not a total loss. Take the set of natural numbers...Ch. 3.2 - Prob. 30MS Ch. 3.2 - Piles of peanuts (ExH). You have infinitely many...Ch. 3.2 - The big city (S). Not-Finite City (also known as...Ch. 3.2 - Dont lose your marbles. Suppose you have...Ch. 3.2 - Make a guess. Guess an infinite set that does not...Ch. 3.2 - Coloring. Consider the infinite collection of...Ch. 3.2 - Ping-Pong balls on parade (H). This Mindscape is...Ch. 3.2 - Primes. Show that the set of all prime numbers has...Ch. 3.2 - A grand union. Suppose you have two sets, and each...Ch. 3.2 - Unnoticeable pruning. Suppose you have any...Ch. 3.2 - Pink ping pong possibilities. You have a box...Ch. 3.2 - Plot the dots (H). The table below gives a...Ch. 3.2 - 1 to 1 or not 1 to 1? Does the table below give a...Ch. 3.2 - Roommates. Your school has 4000 students who want...Ch. 3.3 - Shake em up. What did Georg Cantor do that shook...Ch. 3.3 - Detecting digits. Heres a list of three numbers...Ch. 3.3 - Delving into digits. Consider the real number...Ch. 3.3 - Undercover friend (ExH). Your friend gives you a...Ch. 3.3 - Underhanded friend. Now you friend shows, you a...Ch. 3.3 - Dodgeball. Revisit the game of Dodgeball from...Ch. 3.3 - Dont dodge the connection (S). Explain the...Ch. 3.3 - Cantor with 3s and 7s. Rework Cantors proof from...Ch. 3.3 - Cantor with 4s and 8s. Rework Cantors proof from...Ch. 3.3 - Think positive. Prove that the cardinality of the...Ch. 3.3 - Diagonalization. Cantors proof is often referred...Ch. 3.3 - Digging through diagonals. First, consider the...Ch. 3.3 - Coloring revisited (ExH). In Mindscape 35 of the...Ch. 3.3 - Prob. 14MS Ch. 3.3 - The first digit (H). Suppose that, in constructing...Ch. 3.3 - Ones and twos (H). Show that the set of all real...Ch. 3.3 - Pairs (S). In Cantors argument, is it possible to...Ch. 3.3 - Three missing. Given a list of real numbers, as in...Ch. 3.3 - Prob. 19MS Ch. 3.3 - Prob. 20MS Ch. 3.3 - Nines. Would Cantors argument work if we used 2...Ch. 3.3 - Missing irrational. Could you modify the...Ch. 3.3 - Logging cardinality. The function graphed here is...Ch. 3.3 - U-graph it. Using a graphic or on-line calculator,...Ch. 3.3 - Is a square a one-to-one correspondence? (H)...Ch. 3.3 - Is a cube a one-to-one correspondence? Sketch a...Ch. 3.3 - Find the digit. Your friend is thinking of a real...Ch. 3.4 - Prob. 1MS Ch. 3.4 - Power play. Define the power set of a given set.Ch. 3.4 - Prob. 3MS Ch. 3.4 - Prob. 4MS Ch. 3.4 - Solar power. What is the cardinality of the power...Ch. 3.4 - All in the family (ExH). A family of four tries to...Ch. 3.4 - Making an agenda (H). There are eight members on...Ch. 3.4 - The power of sets (S). Let S={ !,@,#,$,%, }. Below...Ch. 3.4 - Prob. 9MS Ch. 3.4 - Identifying the power. Let S be the set given by...Ch. 3.4 - Prob. 11MS Ch. 3.4 - Another two. Suppose S is the set defined by S={...Ch. 3.4 - Prob. 13MS Ch. 3.4 - Finite Cantor (H). Suppose that S is the set...Ch. 3.4 - One real big set. Describe (in words) a set whose...Ch. 3.4 - Prob. 16MS Ch. 3.4 - The Ultra Grand Hotel (S). Could there be an...Ch. 3.4 - Prob. 18MS Ch. 3.4 - Prob. 19MS Ch. 3.4 - The number name paradox. Let S be the set of all...Ch. 3.4 - Adding another. Suppose that you have any infinite...Ch. 3.4 - Ones and twos. Describe a one-to-one...Ch. 3.4 - Enjoying the exponential function. Consider the...Ch. 3.4 - Prob. 28MS Ch. 3.4 - Power play. Simplify the following expressions:...Ch. 3.4 - Powerful products. For each funciton given below,...Ch. 3.4 - Generalizing equality. Throughout this chapter we...Ch. 3.5 - Lining up. Can you draw a line segment that has...Ch. 3.5 - Reading between the lines. Use the figure below to...Ch. 3.5 - De line and Descartes. Put line segments L and M...Ch. 3.5 - Red line rendezvous (H). Given the equation for...Ch. 3.5 - Rendezvous two. Given the equation for the red...Ch. 3.5 - A circle is a cirde (H). Prove that a small circle...Ch. 3.5 - A circle is a square. Prove that a small circle...Ch. 3.5 - A circle is a triangle. Prove that a small circle...Ch. 3.5 - Stereo connections (ExH). Given the stereogiaphic...Ch. 3.5 - More stereo connections. Given the stereographic...Ch. 3.5 - Perfect shuffle problems (H). Suppose we used our...Ch. 3.5 - More perfect shuffle problems. Suppose we used our...Ch. 3.5 - Gouping digits. Given the grouping of digits...Ch. 3.5 - Where it came from. Given the grouping of digits...Ch. 3.5 - Group fix (S). Consider the point on the line from...Ch. 3.5 - Is there more to a cube? Prove that the...Ch. 3.5 - T and L (H). Prove that the cardinalities of...Ch. 3.5 - Infinitely long is long. Must it be the case that...Ch. 3.5 - Plugging up the north pole (ExH). What would...Ch. 3.5 - 3D stereo (S). Let S be the set of points on the...Ch. 3.5 - Stereo images. Given your answer to the preceding...Ch. 3.5 - Ground shuffle. Carefully verify that the pairing...Ch. 3.5 - Giving the rolled-up interval a tan. The graph...Ch. 3.5 - Back and forth. The function y=5x2 gives a...Ch. 3.5 - Forth and back. The function y=3x+1 gives a...Ch. 3.5 - Lining up (H). Find a function that gives a...Ch. 3.5 - Queuing up. Find a function that gives a...

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