Staircase Light Control Logic Explained

The question describes a common scenario for a staircase light, where the light is controlled by two switches, one at the top and one at the bottom. The key condition given is that the light is ON when and only when one of the switches is ON and the other switch is OFF. This behavior is characteristic of an Exclusive OR (XOR) logic gate.

Let's define our variables for the two switches and the staircase light:

• Let A represent the state of the first switch (e.g., at the top of the stairs).
• Let B represent the state of the second switch (e.g., at the bottom of the stairs).
• Let L represent the state of the staircase light.

We can assign '1' for ON and '0' for OFF for the switches and the light. The condition "light is ON when and only when one of the switches is ON and the other switch is OFF" translates to the following scenarios:

• If switch A is OFF (0) and switch B is ON (1), the light L is ON (1).
• If switch A is ON (1) and switch B is OFF (0), the light L is ON (1).
• If both switches are OFF (0, 0), the light L is OFF (0).
• If both switches are ON (1, 1), the light L is OFF (0).

Staircase Light Truth Table Analysis

To better visualize this logic and derive the logic equation in SOP form, we can construct a truth table showing all possible combinations of the two switches' states (A and B) and the resulting state of the staircase light (L).

Logic Equation Derivation in SOP Form

The Sum of Products (SOP) form of a Boolean expression is obtained by ORing (summing) the product terms for which the output is '1' in the truth table. From our truth table, the staircase light L is ON (output is '1') in two specific cases:

• Case 1: Switch A is OFF (0) and Switch B is ON (1). In Boolean algebra, 'OFF' is represented by the complemented variable (e.g., \(\overline{A}\)) and 'ON' by the uncomplemented variable (e.g., B). The product term for this case is \(\overline{A}B\).
• Case 2: Switch A is ON (1) and Switch B is OFF (0). The product term for this case is \(A\overline{B}\).

To get the final logic equation in SOP form for the staircase light, we sum (OR) these product terms:

\(L = \overline{A}B + A\overline{B}\)

This expression is the standard Boolean equation for an Exclusive OR (XOR) gate, which perfectly matches the described behavior of the staircase light control with two switches.

Logic Equation Options Comparison

Let's compare our derived logic equation with the given options to find the correct SOP form:

• Option 1: \(A + B\) (OR gate) - This would mean the light is ON if A is ON OR B is ON (or both are ON). This does not match our specific condition where only one switch must be ON.
• Option 2: \(\overline {A + B}\) (NOR gate) - This means the light is ON only if both A and B are OFF. This contradicts the condition that one switch must be ON.
• Option 3: \(\overline{A}B + A\overline{B}\) (XOR gate) - This exactly matches our derived SOP form. It correctly represents the scenario where the light is ON when exactly one switch is ON and the other is OFF.
• Option 4: \(AB\) (AND gate) - This would mean the light is ON only if both A and B are ON. This does not match our condition.

Therefore, the logic equation in SOP form that correctly represents the staircase light control is \(\overline{A}B + A\overline{B}\).