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:

Correct Answer:

To design the digital system, we define the output Y as follows:

If the 4-bit input A = A3A2A1A0 (where A3 is MSB, A0 is LSB) represents a number greater than 2 AND is a multiple of 3 → Output Y = 0 (LOGICALLY LOW).
Otherwise (either less than or equal to 2, or not divisible by 3) → Output Y = 1 (LOGICALLY HIGH).
If the input is ≤ 2 → Output is undefined.

We can implement this with combinational logic using a 4-bit input and a truth table to drive the output logic.

Truth Table:

Decimal	Binary	>2?	Mod 3 = 0?	Output Y
0	0000	No	Yes	X (undef)
1	0001	No	No	X (undef)
2	0010	No	No	X (undef)
3	0011	Yes	Yes	0
4	0100	Yes	No	1
5	0101	Yes	No	1
6	0110	Yes	Yes	0
7	0111	Yes	No	1
8	1000	Yes	No	1
9	1001	Yes	Yes	0
10	1010	Yes	No	1
11	1011	Yes	No	1
12	1100	Yes	Yes	0
13	1101	Yes	No	1
14	1110	Yes	No	1
15	1111	Yes	No	1

Explanation (300+ words):

This digital system requires a combinational logic circuit with a 4-bit binary input and a single binary output. The core function of the system is to determine whether a binary input value meets specific conditions and set the output accordingly.

1. Understanding the Input Conditions:

The input is a 4-bit binary number, meaning it can represent values from 0 to 15 (decimal). The system should only define the output if the input value is greater than 2. Therefore, values 0, 1, and 2 are considered undefined and need not be processed in the logic.

2. Multiples of 3:

The system must recognize if the valid input (greater than 2) is a multiple of 3. In the decimal range of 3 to 15, the multiples of 3 are:
3, 6, 9, and 12.

If the input matches one of these values, the output Y must be logically LOW (0). For all other defined values (greater than 2 but not divisible by 3), the output should be logically HIGH (1).

3. Digital Implementation:

To implement this in digital logic:

Use a comparator to check if the input is greater than 2.
Use a modulus detector (e.g., combinational logic that outputs true if a number mod 3 is 0).
Combine these with an AND gate to check if both conditions are met.
Use a NOT gate to invert the result so that the output is low when both conditions are true.

Alternatively, a ROM-based solution or PLD (Programmable Logic Device) can use a lookup table matching the truth table above. This approach is efficient when dealing with small input spaces like a 4-bit system.

Conclusion:

This digital logic system is useful for control applications where actions depend on numeric thresholds and specific divisibility. Designing the output based on arithmetic properties like “greater than” and “multiple of” allows for sophisticated decision-making in digital electronics.