Problem Detector System Logic

The question describes a problem detector system in a factory designed to produce an alarm under specific conditions. It's crucial to understand that only one condition can occur at a time. This implies that the conditions are mutually exclusive. The three conditions are defined as \(q\), \(r\), and \(s\). We need to determine the Boolean output logic for this system.

Understanding Problem Conditions

The system triggers an alarm if any one of the three conditions (\(q\), \(r\), or \(s\)) occurs. The key constraint is that "only one condition can occur at a time." This means we need to consider three distinct scenarios for the alarm to be activated:

• Condition \(q\) occurs: If condition \(q\) is true, then conditions \(r\) and \(s\) must be false.
• Condition \(r\) occurs: If condition \(r\) is true, then conditions \(q\) and \(s\) must be false.
• Condition \(s\) occurs: If condition \(s\) is true, then conditions \(q\) and \(r\) must be false.

Deriving Boolean Expressions for Each Scenario

We can represent these scenarios using Boolean expressions. Let \(q\), \(r\), \(s\) be Boolean variables where '1' indicates the condition is true and '0' indicates it's false. The negation (NOT) of a variable is represented by a bar (e.g., \(\bar{q}\) means 'not \(q\)').

1. When condition \(q\) occurs:

   For the alarm to trigger solely due to \(q\), \(q\) must be true, and \(r\) and \(s\) must be false. This can be expressed as: \(q \text{ AND } \bar{r} \text{ AND } \bar{s}\) In Boolean algebra, this is written as: \($q\bar{r}\bar{s}$\).

2. When condition \(r\) occurs:

   For the alarm to trigger solely due to \(r\), \(r\) must be true, and \(q\) and \(s\) must be false. This can be expressed as: \(\bar{q} \text{ AND } r \text{ AND } \bar{s}\) In Boolean algebra, this is written as: \($\bar{q}r\bar{s}$\).

3. When condition \(s\) occurs:

   For the alarm to trigger solely due to \(s\), \(s\) must be true, and \(q\) and \(r\) must be false. This can be expressed as: \(\bar{q} \text{ AND } \bar{r} \text{ AND } s\) In Boolean algebra, this is written as: \($\bar{q}\bar{r}s$\).

Combining for Overall Output Logic

Since the problem states that the alarm is produced "when one of the three conditions occurs," the overall output logic is the logical OR (sum) of these three mutually exclusive scenarios. If any one of these specific combinations of inputs happens, the alarm should activate.

Therefore, the complete output logic \(Y\) is: \(Y = q\bar{r}\bar{s} + \bar{q}r\bar{s} + \bar{q}\bar{r}s\)

Comparing with Options

Let's compare our derived output logic with the given options. The derived expression is \(q\bar{r}\bar{s} + \bar{q}r\bar{s} + \bar{q}\bar{r}s\).

Looking at the options, we find the following expression:

• qr̅ s̅ + q̅sr̅ + q̅rs̅

Let's check the terms in this option:

All three terms in the given option correspond exactly to the conditions derived from the problem statement, considering the commutative property of Boolean AND operations (e.g., \(\bar{q}s\bar{r}\) is the same as \(\bar{q}\bar{r}s\)). Thus, the option correctly represents the problem's output logic.