Question

Implement the following functions using Shannon's expansion:

a. Implement F=w₁​w₂+w₁w₃+w₂w₃ using only 2 -to-1 MUXs

Answer 1

To implement a Boolean function using only 2-to-1 multiplexers, one applies Shannon's expansion theorem. Here, the function F=w1w2+w1w3+w2w3 is expanded using the theorem, represented with two 2-to-1 MUXs representing the subfunctions for w1=1 and w1=0, and then combined with the final MUX using w1 as the select line.

Step-by-step explanation:

To implement the function F=w1w2+w1w3+w2w3 using only 2-to-1 multiplexers (MUXs), we can use Shannon's expansion theorem. Shannon's theorem states that for any Boolean function F(w1,w2,w3), the function can be expressed as:

F = w1F1(w2, w3) + w1'F0(w2, w3)

where F1 is F with w1 set as 1 and F0 is F with w1 set as 0. In this case:

• F1 = w2 + w3
• F0 = w2w3

We can represent F1 and F0 each with a 2-to-1 MUX:

• MUX1 inputs: 1 (for true), w3, select line: w2. This gives us the output w2 + w3
• MUX2 inputs: 0 (for false), w3, select line: w2. This gives us the output w2w3
  

The final MUX will have select line w1, with input 0 tied to the output of MUX2 (F0) and input 1 tied to the output of MUX1 (F1). This final arrangement yields the original function F when w1 is applied as the select line.