1. **State the problem:** We have two sensors, A and B, controlling a door. The door opens (output 1) if either sensor A is activated or both sensors A and B are activated. 2. **Construct the truth table:** | A | B | Door (Output) | |---|---|--------------| | 0 | 0 | 0 | | 0 | 1 | 0 | | 1 | 0 | 1 | | 1 | 1 | 1 | Explanation: The door opens when A=1 regardless of B, so output is 1 when A=1. 3. **Find the final output equation using SOP method:** From the truth table, the output is 1 for rows where: - A=1, B=0 (row 3) - A=1, B=1 (row 4) The minterms for these rows are: - Row 3: $A \overline{B}$ - Row 4: $AB$ So, the SOP expression is: $$ A \overline{B} + AB $$ 4. **Simplify the expression:** Factor out $A$: $$ A(\overline{B} + B) = A(1) = A $$ So, the final output equation is: $$ \boxed{Y = A} $$ 5. **Logic gate based digital circuit:** Since the output depends only on $A$, the circuit is a direct connection from sensor $A$ to the door control. **Summary:** The door opens if sensor A is activated, regardless of sensor B.