## Step-by-step Solution and Simplification Let's analyze and simplify the given digital logic circuit step by step. --- ### **1. Identify Logic Gates** - **U1, U2:** NOT gates (inverters) - **U3, U4:** AND gates --- ### **2. Assign Inputs** Let: - `A`, `B`, `C` = Inputs (Switches) - LED1 = Output --- ### **3. Trace the Circuit Logic** #### **First Branch (Top Path):** - `A` goes to NOT gate (U1): Output is `A'` - `B` goes directly to U3 - U3 (AND): Inputs = `A'`, `B` - So, output of U3 = `A' • B` #### **Second Branch (Bottom Path):** - `C` goes to NOT gate (U2): Output is `C'` --- #### **Both branches go to an AND gate (U4):** - U4 (AND): Inputs = output of U3 and output of U2 - Inputs are: `(A' • B)` and `C'` - Output = `(A' • B) • C'` --- ### **4. Write the Boolean Expression** \[ \text{LED1} = (A' \cdot B) \cdot C' \] --- ### **5. Simplification** Let's see if it can be further simplified: - The output is **already in minimized Product-of-Terms (AND-OR) form**. - Each logic gate is required by the structure of the expression. #### **Alternative Interpretation** - If you want *fewer* gates, check if you can use NAND or NOR gates for implementation, but the logic can't be reduced further. --- ### **6. Final Answer and Explanation** #### **Boolean Expression:** \[ \boxed{(A' \cdot B \cdot C')} \] #### **Explanation:** - The LED will turn **ON** only if: - `A` is **OFF** (`A'`) - `B` is **ON** - `C` is **OFF** (`C'`) - This is the simplest form for the given logic with the gates provided. --- ## **Summary Table** | A | B | C | LED1 (Output) | |:-:|---|---|---------------| | | 1 | | 1 | | * | * | * | | *(All other combinations)* --- ## **How to Make it Simpler?** - **Replace AND+NOT combinations with NAND gates** if you want to reduce IC count in hardware. - **Logic-wise**, you cannot reduce the expression further; it is already minimal. --- ### **Logic Recap** - The circuit detects when **A = , B = 1, C = ** and lights the LED. - This is a minimal logic circuit for this condition. --- **If you need further simplification in hardware (using fewer chips), consider using universal gates (NAND/NOR). If you want to optimize for logic, this is already the simplest form!**