Please answer Question 2
Assigned questions 1. For each part of this question you will use the following input-output definition table with three inputs 12, 01, 2o and three outputs y2, yi, yo 12 Toy2y1 Yo 1 1 1 0 0 1 1 1 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 (a) (Graded for fair effort completeness ) Draw a logic circuit that corresponds to this input-output table. We recommend the following steps: Draw symbols for the inputs on the left-hand-side and for the outputs on the right-hand side. For each of the three outputs, determine an expression for that output using (some of the inputs 22, 2), 10 and the logic gates XOR, AND, OR, NOT. Draw and label the gates corresponding to the expressions you find, and connect appropri- ately with wires. (b) (Graded for correctness) Does this logic circuit implement the operation of taking the integer r represented in binary fixed-width 3 as (121110)2,3 and computing the following operations so that the result equals (y2yi yo )2,3? If yes, explain why using the definition of binary fixed-width representation and consider all possible inputs. If no, provide specific example input values, use the input-output definition table to compute its outputs for this example, and use the definition to calculate the two numbers being represented and explain why these numbers support your conclusion i. r mod 8 ii. x div 4 (c) (Graded for correctness) Does this logic circuit implement the operation of taking a number represented in width-3 sign-magnitude representation and producing the width-3 sign-magnitude representation of (-1) times this number? If yes, explain why using the definition of sign- magnitude representation and consider all possible inputs. If no, provide specific example input values, use the input-output definition table to compute its outputs for this example, and use the definition to calculate the two numbers being represented and explain why these numbers support your conclusion. (d) (Graded for correctness) Does this logic circuit implement the operation of taking a number represented in width-3 2s complement representation and producing the width-3 2s complement representation of (-1) times this number? If yes, explain why using the definition of 2s com- plement representation and consider all possible inputs. If no, provide specific example input values, use the input-output definition table to compute its outputs for this example, and use the definition to calculate the two numbers being represented and explain why these numbers support your conclusion. (e) (Graded for fair effort completeness) Define an operation on numbers that this logic circuit implements. You may choose whether numbers are represented in binary fixed-width 3 or in width-3 sign magnitude or in width-3 2s complement. Use precise mathematical language to describe your operation and explain why the outputs of the circuit match the operation for all possible inputs. 2. For this question we will consider the logic circuit for the "less than relationship (a) (Graded for fair effort completeness) Write the input-output definition table for a logic circuit with inputs 20,40, 21, yi representing (2160)2,2 and (y1 yo 2.2 and output z such that 2 = 1 exactly when (C170)2,2 (4190)2,2 . Hint: Your table will need 16 rows of input values. Consider how to organize them. (b) (Graded for fair effort completeness) Write a compound proposition that is logically equivalent to the expression for z using only the propositional connectives A, V, and -. Hint: CNF and DNF may be helpful here. (c) (Graded for correctness) Show that -((21 y) A (20 yo)) is not logically equivalent to z by giving a specific example for the values of 20, yo, 21, y where z is different from the value given by this compound proposition. Use the definition of z and the definition of propositional connectives and logical equivalence to support your conclusion. (d) (Graded for correctness) Show that the collection of compound propositions that has z and -( (21+ y) (20 + y)) is consistent by giving a specific example for the values of xo, yo, 11, y1 where z and this com- pound proposition both evaluate to T. Use the definition of z and the definition of propositional connectives and consistency to support your conclusion. 3. (Graded for fair effort completeness) Choose a paragraph (2-5 sentences) from a book / news article/ manual and include it as part of your submission (include citation information for your source). Define relevant meanings for propositional variables and use these to translate each sentence in the paragraph so that the logical structure is expressed using propositional connectives. Determine whether the collection of compound propositions representing the sentences in the paragraph is consistent, and explain your answer. Assigned questions 1. For each part of this question you will use the following input-output definition table with three inputs 12, 01, 2o and three outputs y2, yi, yo 12 Toy2y1 Yo 1 1 1 0 0 1 1 1 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 (a) (Graded for fair effort completeness ) Draw a logic circuit that corresponds to this input-output table. We recommend the following steps: Draw symbols for the inputs on the left-hand-side and for the outputs on the right-hand side. For each of the three outputs, determine an expression for that output using (some of the inputs 22, 2), 10 and the logic gates XOR, AND, OR, NOT. Draw and label the gates corresponding to the expressions you find, and connect appropri- ately with wires. (b) (Graded for correctness) Does this logic circuit implement the operation of taking the integer r represented in binary fixed-width 3 as (121110)2,3 and computing the following operations so that the result equals (y2yi yo )2,3? If yes, explain why using the definition of binary fixed-width representation and consider all possible inputs. If no, provide specific example input values, use the input-output definition table to compute its outputs for this example, and use the definition to calculate the two numbers being represented and explain why these numbers support your conclusion i. r mod 8 ii. x div 4 (c) (Graded for correctness) Does this logic circuit implement the operation of taking a number represented in width-3 sign-magnitude representation and producing the width-3 sign-magnitude representation of (-1) times this number? If yes, explain why using the definition of sign- magnitude representation and consider all possible inputs. If no, provide specific example input values, use the input-output definition table to compute its outputs for this example, and use the definition to calculate the two numbers being represented and explain why these numbers support your conclusion. (d) (Graded for correctness) Does this logic circuit implement the operation of taking a number represented in width-3 2s complement representation and producing the width-3 2s complement representation of (-1) times this number? If yes, explain why using the definition of 2s com- plement representation and consider all possible inputs. If no, provide specific example input values, use the input-output definition table to compute its outputs for this example, and use the definition to calculate the two numbers being represented and explain why these numbers support your conclusion. (e) (Graded for fair effort completeness) Define an operation on numbers that this logic circuit implements. You may choose whether numbers are represented in binary fixed-width 3 or in width-3 sign magnitude or in width-3 2s complement. Use precise mathematical language to describe your operation and explain why the outputs of the circuit match the operation for all possible inputs. 2. For this question we will consider the logic circuit for the "less than relationship (a) (Graded for fair effort completeness) Write the input-output definition table for a logic circuit with inputs 20,40, 21, yi representing (2160)2,2 and (y1 yo 2.2 and output z such that 2 = 1 exactly when (C170)2,2 (4190)2,2 . Hint: Your table will need 16 rows of input values. Consider how to organize them. (b) (Graded for fair effort completeness) Write a compound proposition that is logically equivalent to the expression for z using only the propositional connectives A, V, and -. Hint: CNF and DNF may be helpful here. (c) (Graded for correctness) Show that -((21 y) A (20 yo)) is not logically equivalent to z by giving a specific example for the values of 20, yo, 21, y where z is different from the value given by this compound proposition. Use the definition of z and the definition of propositional connectives and logical equivalence to support your conclusion. (d) (Graded for correctness) Show that the collection of compound propositions that has z and -( (21+ y) (20 + y)) is consistent by giving a specific example for the values of xo, yo, 11, y1 where z and this com- pound proposition both evaluate to T. Use the definition of z and the definition of propositional connectives and consistency to support your conclusion. 3. (Graded for fair effort completeness) Choose a paragraph (2-5 sentences) from a book / news article/ manual and include it as part of your submission (include citation information for your source). Define relevant meanings for propositional variables and use these to translate each sentence in the paragraph so that the logical structure is expressed using propositional connectives. Determine whether the collection of compound propositions representing the sentences in the paragraph is consistent, and explain your
