Question

An automated system has been designed for supplying water in a particular time period daily to manage usage of water without wasting in a particular area.

After the specified period of time, the main valve of the house automatically stops. If not, the water meter will be able to operate because of the air pressure of the main pipeline.

Three automated sensors have been introduced to automatically activate the pump to supply water during that time of day to a house.

Water Volume Sensor (P) mounted in the tank and once the tank reaches the maximum water level, the sensor will operate. The logical value of it is 0 and then closes the valve automatically.

When there is not enough water in the tank, the water volume sensor should be turned off and the logical value of is 1. When there is not enough water in the tank, the main valve that receives the water is automatically opened.

The sensor is operated at the time the water is processed by the time sensor (Q) and the logical value is 1.

A water sensor (R) has been installed to detect the release of water in the main water pump to the home. The sensor is activated when residences s use water and the logical value is 1.

When there is not enough water in the tank, the main water valve for the house is automatically opened. But that time should be the watering time. Even when the tank has enough water, the main water valve for the home is automatically opened when residents use the water.



(i) Build a truth table that automatically opens / doesn't open the main water valve to the home

(ii)Build the Boolean expression in SOP form for the opening of the main valve automatically.

Answer 1

The truth table for the main valve opening is attached below. The Boolean expression in SOP form is Main Valve Open = (P OR (Q AND NOT P)) OR R.

To fill the truth table for the main valve opening, we follow these steps:

Step 1: Analyze the problem:

I identified the three sensors (P, Q, R) and their conditions for activation.

I understood the main valve opening behavior based on tank level, watering time, and water usage.

2. Define the variables and their values:

P: Tank level (0 - full, 1 - empty)

Q: Watering time (0 - not watering, 1 - watering)

R: Water usage (0 - no usage, 1 - usage)

Main Valve Open: Output (0 - closed, 1 - open)

3. Create the table structure:

List all possible combinations of input values (P, Q, R) in the first three columns.

Add a column for the output (Main Valve Open).

Step 4: Fill the output column based on the rules:

For each combination of input values, analyze the rules for opening the main valve:

If the tank is empty (P = 1), the valve opens regardless of other factors.

If the tank is full (P = 0), the valve opens only if it's watering time (Q = 1) or residents are using water (R = 1).

Based on these rules, assign the output value (0 or 1) for each combination.

Step 5: Verification:

Double-check the table to ensure all combinations are considered and the valve opening behavior aligns with the given rules.

Boolean Expression in SOP Form:

The main valve opens when:

Tank is empty (P = 1) or

It's watering time and the tank is full (Q = 1 AND P = 0) or

Residents are using water (R = 1)

Therefore, the Boolean expression in SOP form is:

Main Valve Open = (P OR (Q AND NOT P)) OR R

This expression can be simplified further using DeMorgan's Law:

Main Valve Open = P + (Q * NOT P) + R

Main Valve Open = P + Q * P' + R

Main Valve Open = P + QR + R

This is the simplest SOP form for the main valve opening condition.

This truth table and expression assume that the sensors and valves function correctly. In a real-world system, additional logic and safety measures might be necessary.