Let's solve how to **implement a Full Adder using as few 4-to-1 multiplexers (MUXes) as possible** based on the image. ### 1. **Full Adder Truth Table** | X | Y | Cin | Sum (S) | Cout | |---|---|-----|---------|------| | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 1 | 0 | | 0 | 1 | 0 | 1 | 0 | | 0 | 1 | 1 | 0 | 1 | | 1 | 0 | 0 | 1 | 0 | | 1 | 0 | 1 | 0 | 1 | | 1 | 1 | 0 | 0 | 1 | | 1 | 1 | 1 | 1 | 1 | ### 2. **Using 4-to-1 MUXes** A 4-to-1 MUX can implement any function of 2 select lines and 4 data inputs: - Select: S1, S0 - Inputs: I0, I1, I2, I3 #### Let’s choose selectors: Let’s use **X** and **Y** as the select lines for both MUXes. #### a. **SUM Output Implementation** For a Full Adder: - S = X ⊕ Y ⊕ Cin Let's write S in terms of X and Y (selects) and Cin (data): For S = f(X, Y, Cin): | X | Y | S (when Cin = 0) | S (when Cin = 1) | |---|---|------------------|------------------| | 0 | 0 | 0 | 1 | | 0 | 1 | 1 | 0 | | 1 | 0 | 1 | 0 | | 1 | 1 | 0 | 1 | So for each (X,Y) = (select lines), S = Cin if X⊕Y=1, else S = ¬Cin. So, for each combination: - (0,0): S = Cin - (0,1): S = ¬Cin - (1,0): S = ¬Cin - (1,1): S = Cin So, the truth table for the MUX inputs: - I0 = Cin - I1 = ¬Cin - I2 = ¬Cin - I3 = Cin **You need NOT gates for ¬Cin, which can be done externally or with extra MUXes if you are restricted to using only MUXes.** #### b. **Cout Output Implementation** Cout = majority(X, Y, Cin) = XY + YCin + CinX Let’s fill the table for Cout: | X | Y | Cin | Cout | |---|---|-----|------| | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 0 | | 0 | 1 | 0 | 0 | | 0 | 1 | 1 | 1 | | 1 | 0 | 0 | 0 | | 1 | 0 | 1 | 1 | | 1 | 1 | 0 | 1 | | 1 | 1 | 1 | 1 | Let’s use X and Y as selectors: | X | Y | Cout (Cin=0) | Cout (Cin=1) | |---|---|--------------|--------------| | 0 | 0 | 0 | 0 | | 0 | 1 | 0 | 1 | | 1 | 0 | 0 | 1 | | 1 | 1 | 1 | 1 | So, the MUX inputs (based on Cin): - (0,0): Cout = 0 (always 0) - (0,1): Cout = Cin - (1,0): Cout = Cin - (1,1): Cout = 1 (always 1) So, the MUX inputs: - I0 = 0 - I1 = Cin - I2 = Cin - I3 = 1 --- ### **Final Implementation** - **Sum (S):** 1 MUX, inputs: Cin, ¬Cin, ¬Cin, Cin; Select: X,Y - **Cout:** 1 MUX, inputs: 0, Cin, Cin, 1; Select: X,Y #### **You need two 4-to-1 MUXes** to implement a Full Adder. ### **Summary Table** | Output | MUX Select | I0 | I1 | I2 | I3 | |--------|------------|------|-------|-------|------| | Sum | X, Y | Cin | ¬Cin | ¬Cin | Cin | | Cout | X, Y | 0 | Cin | Cin | 1 | **As few as possible:** **You need only 2 (two) 4-to-1 MUXes** to implement a Full Adder. --- **Let me know if you need a schematic drawing!**

Certainly! Here's a schematic diagram of implementing a Full Adder using two 4-to-1 multiplexers (MUXes): --- ### **Full Adder Implementation Using Two 4-to-1 MUXes** #### **Components:** - 2 × 4-to-1 MUXes - NOT gate (to generate ¬Cin) --- ### **Connections** #### **1. Generating ¬Cin** - Input: Cin - Output: ¬Cin (via a NOT gate) --- ### **MUX 1: Sum (S) Calculation** - **Select lines:** X (S1), Y (S0) - **Inputs:** - I0 = Cin - I1 = ¬Cin - I2 = ¬Cin - I3 = Cin - **Output:** Sum (S) **Operation:** - When X=0, Y=0: select I0 → S=Cin - When X=0, Y=1: select I1 → S=¬Cin - When X=1, Y=0: select I2 → S=¬Cin - When X=1, Y=1: select I3 → S=Cin --- ### **MUX 2: Carry-Out (Cout) Calculation** - **Select lines:** X (S1), Y (S0) - **Inputs:** - I0 = 0 - I1 = Cin - I2 = Cin - I3 = 1 - **Output:** Cout **Operation:** - When X=0, Y=0: select I0 → Cout=0 - When X=0, Y=1: select I1 → Cout=Cin - When X=1, Y=0: select I2 → Cout=Cin - When X=1, Y=1: select I3 → Cout=1 --- ### **Schematic Diagram** ``` X ---------+--------------+ | | | | | [MUX1] --- Sum (S) | | Y ---------+--------------+ | | | [MUX2] --- Carry Out (Cout) | Cin ----+-----------+-- NOT ---- ¬Cin ``` - The **NOT gate** takes Cin and outputs ¬Cin. - **MUX1** (Sum): Select lines X, Y; Inputs as above. - **MUX2** (Cout): Select lines X, Y; Inputs as above. --- ### **Summary:** - **Total MUXes:** 2 - **Additional gates:** 1 NOT gate for ¬Cin - **Inputs:** X, Y, Cin - **Outputs:** Sum (S), Carry out (Cout) --- **Note:** If you want to implement only with MUXes, you can generate ¬Cin using another MUX or an inverter if available, but typically, a single NOT gate suffices. Would you like a graphical diagram with all connections drawn out in detail?