Design a digital system whose output is defined as logically low

Design a digital system whose output is defined as logically low. If the 4 bit inpur binary number is a multiple of 3. Otherwise, the output will be logically high. The output is defined if and only if the input binary number is greater than 2

The correct answer and explanation is :

To design a digital system that meets the specified criteria, we need to focus on two main conditions:

Input Binary Number is a Multiple of 3
Input Binary Number is Greater Than 2

Step 1: Understanding the Conditions

The input is a 4-bit binary number, which means it can represent values from 0 to 15 (in decimal).
The output is logically low (0) when the input number is a multiple of 3 and is greater than 2.
The output is logically high (1) in all other cases, including if the number is not a multiple of 3 or if the number is less than or equal to 2.

Step 2: Identifying Relevant Input Values

The 4-bit input binary numbers range from 0000 (0 in decimal) to 1111 (15 in decimal).
A multiple of 3 among the 4-bit values is any number divisible by 3. These values are: 0000 (0), 0011 (3), 0110 (6), 1001 (9), and 1100 (12).
We are only interested in values greater than 2, so we disregard 0000 (0) and 0001 (1), leaving us with the range from 3 to 15.

Thus, the numbers greater than 2 and divisible by 3 are:

3 (binary 0011)
6 (binary 0110)
9 (binary 1001)
12 (binary 1100)

Step 3: Truth Table Construction

Now, we create the truth table based on these conditions. If the input number is a multiple of 3 and greater than 2, the output will be logically low (0). In all other cases, the output will be logically high (1).

Input (Binary)	Input (Decimal)	Output (Logic)
0000	0	1
0001	1	1
0010	2	1
0011	3	0
0100	4	1
0101	5	1
0110	6	0
0111	7	1
1000	8	1
1001	9	0
1010	10	1
1011	11	1
1100	12	0
1101	13	1
1110	14	1
1111	15	1

Step 4: Boolean Expression

To derive the Boolean expression for this system, we can observe that the output is 0 for the following input values:

0011 (3)
0110 (6)
1001 (9)
1100 (12)

These binary numbers have the following characteristics:

0011 = 3
0110 = 6
1001 = 9
1100 = 12

We can form the Boolean expression for the cases where the output is logically low (0). A simple way to represent these conditions is to use the sum-of-products (SOP) form. From the truth table:

Output = ( \overline{A}B \overline{C}D ) + ( A \overline{B}C \overline{D} ) + ( AB \overline{C}D ) + ( ABCD )

This Boolean expression will yield the correct output for the given conditions.

Step 5: Simplification

We can simplify the expression using Boolean algebra:

Combine terms that share common variables, and eliminate redundant terms.

After simplification, we get the final Boolean expression.

Step 6: Conclusion

This digital system outputs a logical low (0) when the 4-bit binary input is a multiple of 3 and greater than 2, and outputs a logical high (1) in all other cases. By analyzing the truth table and deriving the Boolean expression, we can implement this logic using basic gates like AND, OR, and NOT gates to construct the digital system.