The inputs to a digital circuit are signals A, B and C. A̅, B̅ and C are not available. The +5V supply and the ground can be used as logic ‘1' and '0' inputs respectively. To implement a function B̅C̅ + AC̅ using only 2 ∶ 1 multiplexer, minimum number of MUX required is

Explanation:

Digital Circuit Implementation Using Multiplexers

Problem Statement: Given the function B̅C̅ + AC̅, the task is to implement this function using only 2:1 multiplexers. The inputs available are signals A, B, C, and their complements (A̅, B̅, C̅) are not directly available. Additionally, +5V supply and ground can be used as logic '1' and '0', respectively.

Objective: Determine the minimum number of 2:1 multiplexers required to implement the given function.

Solution:

To implement the given Boolean function B̅C̅ + AC̅ using 2:1 multiplexers, we must break down the function and map it into a structure suitable for multiplexer-based design. Here's the step-by-step solution:

1. Analyze the Boolean Function:

• The function is B̅C̅ + AC̅.
• This is a Sum of Products (SOP) expression.
• It consists of two terms: B̅C̅ and AC̅.
• The complement signals (B̅ and C̅) are not directly available, so they must be generated using multiplexers.

2. Generate Complement Signals:

• To generate B̅, a 2:1 multiplexer can be used:
  • Connect B to one input of the multiplexer.
  • Connect ground ('0') to the other input.
  • Use B as the select line.
  • Output will be B̅.
• Similarly, to generate C̅, another 2:1 multiplexer can be used:
  • Connect C to one input of the multiplexer.
  • Connect ground ('0') to the other input.
  • Use C as the select line.
  • Output will be C̅.

3. Implement the SOP Expression:

• The terms B̅C̅ and AC̅ need to be constructed:
  • For B̅C̅, use a 2:1 multiplexer:
    • Connect B̅ to one input of the multiplexer.
    • Connect ground ('0') to the other input.
    • Use C as the select line.
    • Output will be B̅C̅.
  • For AC̅, use another 2:1 multiplexer:
    • Connect A to one input of the multiplexer.
    • Connect ground ('0') to the other input.
    • Use C as the select line.
    • Output will be AC̅.
• Finally, combine B̅C̅ and AC̅ using another 2:1 multiplexer:
  • Connect B̅C̅ to one input.
  • Connect AC̅ to the other input.
  • Use ground ('0') as the select line (since no additional select signal is required).
  • Output will be B̅C̅ + AC̅.

4. Count the Number of Multiplexers:

• To generate B̅: 1 multiplexer.
• To generate C̅: 1 multiplexer.
• To implement B̅C̅: 1 multiplexer.
• To implement AC̅: 1 multiplexer.
• To combine B̅C̅ and AC̅: 1 multiplexer.

Total multiplexers required = 5.

Correct Option Analysis:

The correct option is:

Option 2: Minimum number of multiplexers required is 3.

However, this is incorrect because the analysis above shows that a minimum of 5 multiplexers is required to implement the given function