Boolean Algebra Define the following Operation - IT/Computer Science Assignment Help

Assignment Task:

Task:

Problem 1: (20 points) Let (B,+, •,′, 0, 1) be a Boolean algebra. Define the following operation ∧? (called NAND, a short-hand for “not and”) ???????? ∧? ???????? ? (????????????????)′. a) Give the truth table of ∧?. b) Is ∧? commutative? Associative? Distributive over +? Distributive over •? Prove your answers c) Express ???????? + ???????? and ???????? • ???????? and ????????’ using the operation ∧? but without using +, • , or ′.

Problem 2: (20 points) a) In a competition judged by a panel of 4 judges, after each player completes his/her presentation, each judge enters his/her vote of yes or no (yes=1, no=0) into a machine. The machine tallies the votes, and returns 1 (that is, pass) if at least 3 judges vote yes. Otherwise, the machine returns 0 (for fail). Express the working of the machine as a Boolean function of four variables (the 4 judges’ votes). b) Same as (a) except that this time the judge #1 has veto power. That is, the machine returns 1 if judge #1 and at least two other judges vote yes; otherwise, the machine returns 0. c) Same as (a) except this time we are interested to see if the 1st and 2nd judges agree and the 3rd and 4th judges disagree. If so, the machine returns 1; otherwise, it returns 0.

Problem 3: (20 points) For each of the three Boolean functions you obtained in problem 2, put the function in disjunctive normal form (DNF) and in conjunctive normal form (CNF). Problem 4: (20 points) For each of the following Boolean expressions, give the truth table, and put the expression in DNF and CNF. a) ???????????????????????? + ????????’(???????? + ????????′????????′) b) ????????(???????? + ????????????????’) + ????????’????????’ c) ????????(???????????????? + ????????????????′???????? + ????????′????????????????) d) (???????????????? + ????????′????????′ )(???????????????? + ????????′????????′) + ???????????????????????? Problem 5: (20 points) Minimize each of the expressions of problem 4 using Karnaugh maps. Show the Karnaugh maps. Bonus Problem: (5 points) a) Let ????????,???????? and ???????? be three Boolean variables. Also, let ????????(????????,????????, ????????) = ? 0 if the number of variables of value 1 is even 1 if the number of variables of value 1 is odd and ????????(????????,????????, ????????) = ? 0 if the number of variables of value 1 is at most 1 1 if the number of variables of value 1 is at least

2 . Express ???????? and ???????? as Boolean expressions in terms of ????????,???????? and ????????. b) Design a circuit for ???????? made up of no more than 9 and/or-gates, and a circuit for ???????? made up of no more than 5 and/or-gates. Note that you can use inverters and they don’t count towards the number of gates used.

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