This computer science problem involves algorithmic thinking and programming concepts. The solution below explains the approach, logic, and implementation step by step.
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Here are the answers for questions 2, 3, and 4: 2. a) Explain the three Boolean algebra laws you know using ORing and ANDing system. 1. Commutative Law: This law states that the order of operands does not affect the result of an operation. For ORing: A + B = B + A For ANDing: A · B = B · A 2. Associative Law: This law states that the grouping of operands does not affect the result of an operation when performing the same operation multiple times. For ORing: (A + B) + C = A + (B + C) For ANDing: (A · B) · C = A · (B · C) 3. Distributive Law: This law describes how operations interact with each other. It allows for the expansion or factorization of Boolean expressions. A · (B + C) = (A · B) + (A · C) A + (B · C) = (A + B) · (A + C) b) Explain logic gate in terms of number system. A logic gate operates within the binary number system, which uses only two digits: 0 and 1. These digits represent the two possible states of an electrical signal (e.g., 0 for low voltage/off, 1 for high voltage/on). Logic gates take one or more binary inputs and produce a single binary output based on a specific Boolean function, effectively performing arithmetic or logical operations on these binary numbers. 3. Simplify the following Boolean algebra. i) A + AB + A + B Step 1: Apply the Idempotence Law (X+X=X) for A+A. A + AB + A + B = A + AB + B Step 2: Apply the Absorption Law (X + XY = X+Y). A + AB + B = A + B The simplified expression is A+B. ii) Z = AB + A(B+C) + B(B+C) Step 1: Apply the Distributive Law (X(Y+Z) = XY+XZ). Z = AB + AB + AC + BB + BC Step 2: Apply the Idempotence Law (X+X=X and XX=X). Z = AB + AC + B + BC Step 3: Factor out B from terms containing B. Z = B(A+1) + AC + BC Step 4: Apply the Identity Law (X+1=1). Z = B(1) + AC + BC Z = B + AC + BC Step 5: Factor out B again from terms containing B. Z = B(1+C) + AC Step 6: Apply the Identity Law (1+X=1). Z = B(1) + AC Z = B + AC The simplified expression is B+AC. iii) F = A + ABC + ABC Step 1: Factor out AC from the last two terms. F = A + AC(B + B) Step 2: Apply the Complement Law (B + B = 1). F = A + AC(1) Step 3: Apply the Identity Law (X · 1 = X). F = A + AC Step 4: Apply the Absorption Law (X + XY = X+Y). F = A + C The simplified expression is A+C. 4. Form the truth table using the Boolean expression of the following: i) F = A + ABC + ABC From 3(iii), we simplified this expression to F = A+C. We will use this simplified form to construct the truth table. | A | C | F = A+C | |---|---|---------| | 0 | 0 | 0 | | 0 | 1 | 1 | | 1 | 0 | 1 | | 1 | 1 | 1 | ii) Z = AB + A(B+C) + B(B+C) From 3(ii), we simplified this expression to Z = B+AC. We will use this simplified form to construct the truth table. | A | B | C | AC | Z = B+AC | |---|---|---|----|----------| | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 0 | 0 | | 0 | 1 | 0 | 0 | 1 | | 0 | 1 | 1 | 0 | 1 | | 1 | 0 | 0 | 0 | 0 | | 1 | 0 | 1 | 1 | 1 | | 1 | 1 | 0 | 0 | 1 | | 1 | 1 | 1 | 1 | 1 | Drop the next question.