2. Construct a combinatorial circuit using inverters, OR gates, and AND gates that produces the output. You must label the input(s) and the output for each one of the gates. Simply writing/labeling the final output will result in 0 credit awarded. (15 points) $$ -(p \wedge(q \vee-r)) $$

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1. **Inverting $$r$$:**  
   - **Gate:** Inverter  
   - **Input:** $$r$$  
   - **Output:** Let $$r' = \neg r$$

2. **Computing $$q \vee \neg r$$:**  
   - **Gate:** OR gate  
   - **Inputs:** $$q$$ and $$r'$$  
   - **Output:** Let $$O_1 = q \vee r'$$  
   $$
   O_1 = q \vee \neg r
   $$

3. **Computing $$p \wedge (q \vee \neg r)$$:**  
   - **Gate:** AND gate  
   - **Inputs:** $$p$$ and $$O_1$$  
   - **Output:** Let $$A_1 = p \wedge O_1$$  
   $$
   A_1 = p \wedge (q \vee \neg r)
   $$

4. **Inverting the AND output:**  
   - **Gate:** Inverter  
   - **Input:** $$A_1$$  
   - **Output:** Final output $$F = \neg A_1$$  
   $$
   F = \neg (p \wedge (q \vee \neg r))
   $$

The circuit in block form is as follows:

- **Step 1:**  
  Inverter  
  $$\begin{array}{c}
      \text{Input: } r \
      \text{Output: } r' = \neg r
  \end{array}$$

- **Step 2:**  
  OR gate  
  $$\begin{array}{c}
      \text{Inputs: } q, \; r' \
      \text{Output: } O_1 = q \vee r'
  \end{array}$$

- **Step 3:**  
  AND gate  
  $$\begin{array}{c}
      \text{Inputs: } p, \; O_1 \
      \text{Output: } A_1 = p \wedge O_1
  \end{array}$$

- **Step 4:**  
  Inverter  
  $$\begin{array}{c}
      \text{Input: } A_1 \
      \text{Output: } F = \neg A_1 = \neg (p \wedge (q \vee \neg r))
  \end{array}$$

This completes the construction of the combinatorial circuit using inverters, OR gates, and AND gates.
To construct the combinatorial circuit for $$-(p \wedge (q \vee -r))$$, follow these steps: 1. **Invert $$r$$:** - **Gate:** Inverter - **Input:** $$r$$ - **Output:** $$r'$$ 2. **Compute $$q \vee r'$$:** - **Gate:** OR gate - **Inputs:** $$q$$ and $$r'$$ - **Output:** $$O_1 = q \vee r'$$ 3. **Compute $$p \wedge O_1$$:** - **Gate:** AND gate - **Inputs:** $$p$$ and $$O_1$$ - **Output:** $$A_1 = p \wedge O_1$$ 4. **Invert $$A_1$$:** - **Gate:** Inverter - **Input:** $$A_1$$ - **Output:** $$F = \neg A_1 = \neg (p \wedge (q \vee \neg r))$$ **Circuit Summary:** - **Inverter:** $$r \rightarrow r'$$ - **OR gate:** $$q \vee r' \rightarrow O_1$$ - **AND gate:** $$p \wedge O_1 \rightarrow A_1$$ - **Inverter:** $$A_1 \rightarrow F$$ This circuit correctly implements the logical expression $$-(p \wedge (q \vee -r))$$ using inverters, OR gates, and AND gates.

Construct a combinatorial circuit using inverters, OR gates, and AND gates that produces the output. You must
label the input(s) and the output for each one of the gates. Simply writing/labeling the final output will result in 0
credit awarded. (15 points)

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