Question

Computer Science

Posted 6 months ago

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ou are supposed to implement a circuit that takes three inputs A, B, and C. When the values of the input A, B, and C, interpreted as an unsigned integer is divisible by 3, then the circuit should output 1. For example, if A is 0, B is 1, and C is 1, then the unsigned integer value is 011, which is 3 and is divisible by 3. Write the minimal Boolean expression for the circuit.
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Answer from Sia

Posted 6 months ago

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To solve the problem of creating a circuit that outputs 1 when the unsigned integer value of inputs A, B, and C is divisible by 3, we need to consider all possible combinations of A, B, and C and their corresponding outputs. Since A, B, and C are binary digits, there are $2^3 = 8$ possible combinations. We can create a truth table to determine the output for each combination:
```

A B C | Unsigned Integer | Divisible by 3 | Output

```
From the truth table, we can derive the minimal Boolean expression using Karnaugh maps or Boolean algebra simplification techniques. The minimal Boolean expression that represents the output is:
$F(A, B, C) = \overline{A} \cdot \overline{B} \cdot \overline{C} + \overline{A} \cdot B \cdot C + A \cdot B \cdot \overline{C}$
```

Explanation

This Boolean expression represents the minimal form of the logic needed to output 1 when the binary number ABC is divisible by 3. The overline represents the NOT operation, the dot represents the AND operation, and the plus represents the OR operation.

Step-by-step Instruction

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Determine the output for each combination based on whether the unsigned integer value is divisible by 3

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Use Karnaugh maps or Boolean algebra simplification techniques to derive the minimal Boolean expression

Time Complexity

This concept does not apply to Boolean expression simplification as it is not an algorithm with multiple steps that depend on the size of the input.

Space Complexity

This concept does not apply to Boolean expression simplification as it does not require memory allocation that depends on the size of the input.

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