The given problem is about a 4-bit counting ADC (Analog-to-Digital Converter). The resolution of an ADC refers to the smallest change in the analog input that can be detected and represented by the digital output. In this case, the resolution is given as 0.5V.

To understand the solution, let's break it down step by step:

1. Understanding the ADC:
- An ADC converts analog signals into digital signals by quantizing the analog input voltage.
- The 4-bit ADC in question will have a total of 2^4 = 16 possible digital output values.
- Each bit in the output represents a certain fraction of the full-scale range of the ADC.

2. Determining the full-scale range:
- The resolution of the ADC is given as 0.5V, which means that the full-scale range of the ADC is 2^4 * 0.5V = 8V.
- This means that the ADC can represent analog inputs ranging from 0V to 8V.

3. Determining the digital output for an analog input of 5.8V:
- To determine the digital output for an analog input of 5.8V, we need to find the corresponding digital value within the full-scale range of the ADC.
- Since the full-scale range is 8V, we can calculate the fraction of the full-scale range represented by the analog input as follows: (5.8V / 8V) = 0.725.
- Multiplying this fraction by the total number of digital values (16) gives us the approximate digital output: 0.725 * 16 = 11.6.

4. Rounding the digital output:
- Since the ADC can only provide integer values as the digital output, we need to round the calculated value of 11.6 to the nearest integer.
- Rounding 11.6 to the nearest integer gives us 12.

5. Converting the rounded digital output to binary:
- The rounded digital output of 12 can be represented in binary as 1100.
- Each bit in the binary representation corresponds to a power of 2, starting from the most significant bit (MSB) on the left.
- Therefore, the binary representation 1100 corresponds to the digital output value of 12.

Hence, the correct answer is option 'A' (1100).