Scenario: Imagine you are an engineer tasked with designing the logic for a smart home security system. The system involves various sensors and actuators to monitor and control different aspects of the home's security. Your goal is to use Boolean algebra and logic gates to create a logical framework that efficiently represents the conditions and actions for the smart home security system.

On the basis of the above scenario, answer the following questions.

Question $4$:

a$.$ Determine the equivalence between the simplified Boolean expressions from Q$1$ and the corresponding truth tables from Question $3.$

b$.$ Provide algebraic manipulations and logical reasoning to support your conclusions.

Question $5$:

a$.$ Integrate the simplified Boolean expressions from Q$1$ and the gates from Q$3$ to represent the overall logic for the smart home security system.