Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

8th Edition

ISBN: 9781259676512

Author: Kenneth H Rosen

Publisher: McGraw-Hill Education

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Textbook Question

Chapter 12.1, Problem 23E

Exercises 4-3 deal with the Boolean algebra {0, 1} with addition, multiplication, and complement defined at the ginning of the section. In each case, use a table as in Example 8.

23. Verify the zero property.

The Boolean operator ⊕ , called the XOR operator, is defined by 1 ⊕ 1 = 0 , 1 ⊕ 0 = 1 , 0 ⊕ 1 = 1 , and 0 ⊕ 0 = 0 .

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Chapter 12.1, Problem 22E

Chapter 12.1, Problem 24E

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A function is defined on the interval (-π/2,π/2) by this multipart rule: if -π/2 < x < 0 f(x) = a if x=0 31-tan x +31-cot x if 0 < x < π/2 Here, a and b are constants. Find a and b so that the function f(x) is continuous at x=0. a= b= 3

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. f(x) = (x + 4x4) 5, a = -1 lim f(x) X--1 = lim x+4x X--1 lim X-1 4 x+4x 5 ))" 5 )) by the power law by the sum law lim (x) + lim X--1 4 4x X-1 -(0,00+( Find f(-1). f(-1)=243 lim (x) + -1 +4 35 4 ([ ) lim (x4) 5 x-1 Thus, by the definition of continuity, f is continuous at a = -1. by the multiple constant law by the direct substitution property

Chapter 12 Solutions

Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

Ch. 12.1 - Prob. 1E Ch. 12.1 - Find the values, if any, of the Boolean...Ch. 12.1 - a) Show that(1.1)+(0.1+0)=1 . b) Translate the...Ch. 12.1 - a) Show that(10)+(10)=1 . b) Translate the...Ch. 12.1 - Use a table to express the values of each of these...Ch. 12.1 - Use a table to express the values of each of these...Ch. 12.1 - Use a 3-cubeQ3to represent each of the Boolean...Ch. 12.1 - Use a 3-cubeQ3to represent each of the Boolean...Ch. 12.1 - What values of the Boolean...Ch. 12.1 - How many different Boolean functions are there of...

Ch. 12.1 - Prove the absorption lawx+xy=x using the other...Ch. 12.1 - Show thatF(x,y,z)=xy+xz+yz has the value 1 if and...Ch. 12.1 - Show thatxy+yz+xz=xy+yz+xz .Ch. 12.1 - 3Exercises 14-23 deal the Boolean algebra {0, 1}...Ch. 12.1 - Exercises 14-23 deal with the Boolean algebra {0,...Ch. 12.1 - Prob. 16E Ch. 12.1 - Exercises 14-23 deal with the Boolean algebra {0,...Ch. 12.1 - Prob. 18E Ch. 12.1 - Prob. 19E Ch. 12.1 - Prob. 20E Ch. 12.1 - Prob. 21E Ch. 12.1 - Prob. 22E Ch. 12.1 - Exercises 4-3 deal with the Boolean algebra {0, 1}...Ch. 12.1 - Prob. 24E Ch. 12.1 - Prob. 25E Ch. 12.1 - Prob. 26E Ch. 12.1 - Prove or disprove these equalities. a)x(yz)=(xy)z...Ch. 12.1 - Find the duals of these Boolean expressions. a)x+y...Ch. 12.1 - Prob. 29E Ch. 12.1 - Show that ifFandGare Boolean functions represented...Ch. 12.1 - How many different Boolean functionsF(x,y,z) are...Ch. 12.1 - How many different Boolean functionsF(x,y,z) are...Ch. 12.1 - Show that you obtain De Morgan’s laws for...Ch. 12.1 - Show that you obtain the ab,sorption laws for...Ch. 12.1 - In Exercises 35-42, use the laws in Definition 1...Ch. 12.1 - In Exercises 35-42, use the laws in Definition to...Ch. 12.1 - Prob. 37E Ch. 12.1 - Prob. 38E Ch. 12.1 - In Exercises 35-42, use the laws in Definition 1...Ch. 12.1 - Prob. 40E Ch. 12.1 - Prob. 41E Ch. 12.1 - Prob. 42E Ch. 12.1 - Prob. 43E Ch. 12.2 - Find a Boolean product of the Boolean...Ch. 12.2 - Find the sum of products expansions of these...Ch. 12.2 - Find the sum-of-products expansions of these...Ch. 12.2 - Find the sum-of-products expansions of the Boolean...Ch. 12.2 - Find the sum-of -products expansion of the Boolean...Ch. 12.2 - Find the sum-of-products expansion of the Boolean...Ch. 12.2 - Another way to find a Boolean expression that...Ch. 12.2 - Prob. 8E Ch. 12.2 - Prob. 9E Ch. 12.2 - Another way to find a Boolean expression that...Ch. 12.2 - Prob. 11E Ch. 12.2 - Express each of these Boolean functions using the...Ch. 12.2 - Express each of the Boolean functions in...Ch. 12.2 - Show that a)x=xx . b)xy=(xy)(xy) . c)x+y=(xx)(yy)...Ch. 12.2 - Prob. 15E Ch. 12.2 - Show that{} is functionally complete using...Ch. 12.2 - Express each of the Boolean functions in Exercise...Ch. 12.2 - Express each of the Boolean functions in Exercise...Ch. 12.2 - Show that the set of operators{+,} is not...Ch. 12.2 - Are these sets of operators functionally complete?...Ch. 12.3 - In Exercises 1—5 find the output of the given...Ch. 12.3 - In Exercises 1—5 find the output of the given...Ch. 12.3 - In Exercises 1—5 find the output of the given...Ch. 12.3 - In Exercises 1—5 find the output of the given...Ch. 12.3 - In Exercises 1—5 find the output of the given...Ch. 12.3 - Construct circuits from inverters, AND gates, and...Ch. 12.3 - Design a circuit that implements majority voting...Ch. 12.3 - Design a circuit for a light fixture controlled by...Ch. 12.3 - Show how the sum of two five-bit integers can be...Ch. 12.3 - Construct a circuit for a half subtractor using...Ch. 12.3 - Construct a circuit for a full subtractor using...Ch. 12.3 - Use the circuits from Exercises 10 and 11 to find...Ch. 12.3 - Construct a circuit that compares the two-bit...Ch. 12.3 - Construct a circuit that computes the product of...Ch. 12.3 - Use NAND gates to construct circuits with these...Ch. 12.3 - Use NOR gates to construct circuits for the...Ch. 12.3 - Construct a half adder using NAND gates.Ch. 12.3 - Construct a half adder using NOR gates.Ch. 12.3 - Construct a multiplexer using AND gates, OR gates,...Ch. 12.3 - Find the depth of a) the circuit constructed in...Ch. 12.4 - Prob. 1E Ch. 12.4 - Find the sum-of-products expansions represented by...Ch. 12.4 - Draw the K-maps of these sum-of-products...Ch. 12.4 - Use a K-map to find a minimal expansion as a...Ch. 12.4 - a) Draw a K-map for a function in three variables....Ch. 12.4 - Use K-maps to find simpler circuits with the same...Ch. 12.4 - Prob. 7E Ch. 12.4 - Prob. 8E Ch. 12.4 - Construct a K-map for F(x,y,z) =xz + yz+y z. Use...Ch. 12.4 - Draw the 3-cube Q3 and label each vertex with the...Ch. 12.4 - Prob. 11E Ch. 12.4 - Use a K-map to find a minimal expansion as a...Ch. 12.4 - a) Draw a K-map for a function in four variables....Ch. 12.4 - Use a K-map to find a minimal expansion as a...Ch. 12.4 - Find the cells in a K-map for Boolean functions...Ch. 12.4 - How many cells in a K-map for Boolean functions...Ch. 12.4 - a) How many cells does a K-map in six variables...Ch. 12.4 - Show that cells in a K-map for Boolean functions...Ch. 12.4 - Which rows and which columns of a 4 x 16 map for...Ch. 12.4 - Prob. 20E Ch. 12.4 - Prob. 21E Ch. 12.4 - Use the Quine-McCluskey method to simplify the...Ch. 12.4 - Use the Quine—McCluskey method to simp1i’ the...Ch. 12.4 - Prob. 24E Ch. 12.4 - Use the Quine—McCluskey method to simplify the...Ch. 12.4 - Prob. 26E Ch. 12.4 - Prob. 27E Ch. 12.4 - Prob. 28E Ch. 12.4 - Prob. 29E Ch. 12.4 - Prob. 30E Ch. 12.4 - Prob. 31E Ch. 12.4 - Prob. 32E Ch. 12.4 - show that products of k literals correspond to...Ch. 12 - Define a Boolean function of degreen.Ch. 12 - Prob. 2RQ Ch. 12 - Prob. 3RQ Ch. 12 - Prob. 4RQ Ch. 12 - Prob. 5RQ Ch. 12 - Prob. 6RQ Ch. 12 - Explain how to build a circuit for a light...Ch. 12 - Prob. 8RQ Ch. 12 - Is there a single type of logic gate that can be...Ch. 12 - a) Explain how K-maps can be used to simplify...Ch. 12 - a) Explain how K-maps can be used to simplify...Ch. 12 - a) What is a don’t care condition? b) Explain how...Ch. 12 - a) Explain how to use the Quine-McCluskev method...Ch. 12 - Prob. 1SE Ch. 12 - Prob. 2SE Ch. 12 - Prob. 3SE Ch. 12 - Prob. 4SE Ch. 12 - Prob. 5SE Ch. 12 - Prob. 6SE Ch. 12 - Prob. 7SE Ch. 12 - Prob. 8SE Ch. 12 - Prob. 9SE Ch. 12 - Prob. 10SE Ch. 12 - Prob. 11SE Ch. 12 - Prob. 12SE Ch. 12 - Prob. 13SE Ch. 12 - Prob. 14SE Ch. 12 - Prob. 15SE Ch. 12 - Prob. 16SE Ch. 12 - How many of the 16 Boolean functions in two...Ch. 12 - Prob. 18SE Ch. 12 - Prob. 19SE Ch. 12 - Design a circuit that determines whether three or...Ch. 12 - Prob. 21SE Ch. 12 - A Boolean function that can be represented by a...Ch. 12 - Prob. 23SE Ch. 12 - Prob. 24SE Ch. 12 - Given the values of two Boolean variablesxandy,...Ch. 12 - Prob. 2CP Ch. 12 - Prob. 3CP Ch. 12 - Prob. 4CP Ch. 12 - Prob. 5CP Ch. 12 - Prob. 6CP Ch. 12 - Prob. 7CP Ch. 12 - Prob. 8CP Ch. 12 - Prob. 9CP Ch. 12 - Given the table of values of a Boolean function,...Ch. 12 - Prob. 11CP Ch. 12 - Prob. 12CP Ch. 12 - Prob. 1CAE Ch. 12 - Prob. 2CAE Ch. 12 - Prob. 3CAE Ch. 12 - Prob. 4CAE Ch. 12 - Prob. 5CAE Ch. 12 - Prob. 6CAE Ch. 12 - Prob. 7CAE Ch. 12 - Describe some of the early machines devised to...Ch. 12 - Explain the difference between combinational...Ch. 12 - Prob. 3WP Ch. 12 - Prob. 4WP Ch. 12 - Find out how logic gates are physically...Ch. 12 - Explain howdependency notationcan be used to...Ch. 12 - Describe how multiplexers are used to build...Ch. 12 - Explain the advantages of using threshold gates to...Ch. 12 - Describe the concept ofhazard-free switching...Ch. 12 - Explain how to use K-maps to minimize functions of...Ch. 12 - Prob. 11WP Ch. 12 - Describe what is meant by the functional...

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Solution Summary: The author explains that the zero property is verified using the Boolean algebra left0,1right.

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