Let's break down the step-by-step solution using bubble pushing, so the circuit **only contains inverters, NOR, and NAND gates**. --- ### **Step 1: Identify the Gates** Looking at the circuit: - The top three gates are **OR gates** (curved input side). - The bottom-most gate is an **AND gate** (flat input side). - The final gate (on the right) is an **AND gate**. --- ### **Step 2: Write the Boolean Expression** Let's assign variables: - Top: \( A, B \) - Middle: \( C, D \) - Bottom: \( E, F \) The circuit is: - \( X_1 = A + B \) (OR) - \( X_2 = C + D \) (OR) - \( X_3 = E + F \) (OR) - \( X_4 = X_2 \cdot X_3 \) (AND) - Output = \( X_1 \cdot X_4 \) (AND) So the expression is: \[ (A+B) \cdot ((C+D) \cdot (E+F)) \] --- ### **Step 3: Apply Bubble Pushing** We want to replace OR and AND with NOR, NAND, and inverters only. #### **a) Replace OR gates** An OR gate can be written using NOR (with bubble pushing): \[ A + B = \overline{ \overline{A + B} } \] - The inner part is a **NOR gate**. - The outer bar is an **inverter**. So, each OR gate becomes a NOR gate followed by an inverter. #### **b) Replace AND gates** Similarly, an AND gate can be written using NAND: \[ A \cdot B = \overline{ \overline{A \cdot B} } \] - The inner part is a **NAND gate**. - The outer bar is an **inverter**. So, each AND gate becomes a NAND gate followed by an inverter. --- ### **Step 4: Redraw the Circuit with NOR, NAND, and Inverters** #### **a) Rewrite Each Gate** - **Each OR gate**: Replace with NOR + inverter. - **Each AND gate**: Replace with NAND + inverter. #### **b) Add Bubbles (Push Bubbles)** - A bubble at the output of a gate means negation (inverter). - When you push a bubble from the output to the inputs, you must also invert the gate type (DeMorgan's Law): - \(\overline{A \cdot B} = \overline{A} + \overline{B}\) (NAND becomes NOR with inverted inputs) - \(\overline{A + B} = \overline{A} \cdot \overline{B}\) (NOR becomes NAND with inverted inputs) --- #### **Circuit step-by-step after bubble pushing:** 1. **Replace all OR gates with NOR gates, followed by inverters.** - Each OR output gets a bubble (inverter). 2. **Replace all AND gates with NAND gates, followed by inverters.** - Each AND output gets a bubble (inverter). 3. **Push bubbles towards the inputs as much as possible.** - When two bubbles meet (inverter followed by inverter), they cancel out. --- #### **Final Circuit:** Let's label the outputs of each NOR/NAND gate: - **Top:** \(\overline{A+B}\) (NOR gate) → inverter → \(A+B\) - **Middle:** \(\overline{C+D}\) (NOR gate) → inverter → \(C+D\) - **Bottom:** \(\overline{E+F}\) (NOR gate) → inverter → \(E+F\) - **AND of middle and bottom:** Now, instead of AND, use NAND + inverter: - NAND: \(\overline{(C+D) \cdot (E+F)}\) - Inverter: \((C+D) \cdot (E+F)\) - **Final output:** AND of \(A+B\) and \((C+D) \cdot (E+F)\): - NAND: \(\overline{(A+B) \cdot ((C+D) \cdot (E+F))}\) - Inverter: \((A+B) \cdot ((C+D) \cdot (E+F))\) --- ### **Step 5: Draw the Final Circuit** Here is your circuit, using only NOR, NAND, and inverters: ``` A---| | |>o---| | | | |NOR |---o-----| INV | | | B---|____| | ____| | | | | | | C---| | | |--|NAND|---| INV | |NOR |---| | |_____| | | D---|____| | | | | | | | | E---| | | | | | |NOR |---| | | | F---|____| | | | | | | | | |-------------o------------| | | -----------------------------------------------------o---|NAND|---|INV|---OUTPUT ``` #### **Where and When Bubbles are Added:** - After every NOR or NAND, add an inverter (bubble) at the output. - When connecting to the next gate, if you see two inverters in a row, remove both (they cancel). - Each original OR is now a NOR + inverter. - Each original AND is now a NAND + inverter. --- ### **Conclusion** **Final answer:** By bubble pushing and applying DeMorgan's laws, you can redraw the circuit **using only NOR, NAND, and inverters**. - Replace every OR with a NOR + inverter. - Replace every AND with a NAND + inverter. - Push bubbles (inverters) as far toward the inputs as possible, canceling double inverters when they appear. If you want a hand-drawn version or step-by-step drawing, let me know!