Explanation Multipliers Multipliers in digital electronics are employed with binary digits and more frequently termed as binary multipliers. In brief binary multiplier is an electronic circuit which is used in digital-electronics like in microprocessors and computers to get the result of the product of two binary numbers. They are built by using the logic adders and the logic adders as we know are designed by using the logic gates. To perform various algorithms, they are frequently used in signal processing of digital signals. In the process of multiplication, the number multiplied is known as multiplier and the number to be multiplied is called multiplicand. Binary Multiplication: - The process of multiplication of two binary numbers is same as the multiplication of two decimal numbers. As we know that the binary function only contains 0 and 1 the multiplication is relatively easier as compared to the multiplication of decimal digits. Below are the basic multiplication rules for binary numbers: - 0 × 0 = 0 0 × 1 = 0 1 × 0 = 0 1 × 1 = 1 The result of two binary number multiplication can be achieved by using below methods: - (A.) Partial Product Addition and shifting. (B.) By using parallel multipliers. Let us take an example to understand the multiplication process of two unsigned binary numbers. Let two 4 bit binary numbers as 1010 and 1011 and the multiplication of the two numbers can be calculated as below: - Like decimal numbers, every successive value of partial product must be shifted by one position to the left in relation to the previous value of partial product and it is done before adding all partial product value. Thus, this type of multiplication utilizes the n-shift operation and also adds to obtain the multiplied result of n-shifting operation. To obtain the result of such type of multiplication is known as an array multiplier or a combinational multiplier. Let us take another example to implement a two-bit multiplier. As we know that a two-bit number can hold four numbers from 0 − 3 Now considering the combination if one multiplies 3 × 3 we get 9 which actually requires 4 bits to store this data. So, the multiplier which we design must have 2 ( 2 bit ) numbers of input and output 4 outputs bits. Now consider that the inputs are labelled as A, B, C, D. We will label the outputs W< X, Y, Z where they represent results on the basis of the inputs A, B, C, D. Now we create sum of product equations for all four outputs. As for W we have one case where the output is true, so that we can not further reduce it below one term. W = A B C D Use the below identity to minimize the functions

8th Edition

ISBN: 9781264309689

Author: ROSEN

Publisher: MCG

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Chapter 12 Solutions

DISCRETE MATHEMATICS LOOSELEAF

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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