This problem requires designing a logical circuit to compare two 2-bit numbers $A_1A_0$ and $B_1B_0$, where the two outputs are:

1. $A > B$
2. $A = B$

You need to design the circuit in two ways:

1. Using an 8:1 Multiplexer (MUX).
2. Using a 4-to-16 Decoder.

1. Solution Using an 8:1 MUX

For a 2-bit comparator, the numbers $A = A_1A_0$ and $B = B_1B_0$ can each range from 0 to 3. For the output conditions:

• $A = B$: The values of $A$ and $B$ are the same (00, 01, 10, or 11).
• $A > B$: The values of $A$ and $B$ satisfy $A > B$ (possible for specific combinations).

To use an 8:1 MUX:

• Select lines will be $A_1$, $A_0$, and $B_1$, which cover all combinations of $A$ and $B$.
• Outputs:
  • For $A > B$: Configure the MUX to output 1 for cases where $A > B$ (e.g., $A = 10$ and $B = 01$).
  • For $A = B$: Configure the MUX to output 1 only when $A = B$ for each pair (e.g., both $A$ and $B$ equal 01).

2. Solution Using a 4-to-16 Decoder

With a 4-to-16 decoder:

• Use $A_1, A_0, B_1, B_0$ as the inputs to generate 16 outputs, representing all possible values of $A$ and $B$.
• Outputs:
  • Identify which outputs correspond to $A > B$ and connect them to create a combined output.
  • Similarly, connect outputs for which $A = B$ to form the equality output.

Would you like a step-by-step breakdown of how to implement the MUX and decoder connections?

Follow-Up Questions:

1. How would this circuit design change if we were comparing 3-bit numbers instead of 2-bit numbers?
2. Can we implement this comparator with other logic gates instead of using MUX or Decoder?
3. How would the design change if we also needed an output for $A < B$?
4. What are the advantages of using MUX vs. Decoder in circuit design?
5. How can this logic circuit be optimized for power efficiency?

Tip:

When designing logic circuits, mapping out all input-output possibilities (truth table) can simplify the selection of MUX inputs and outputs.