Question

Explain how to build a circuit for a light controlled by two switches using \({\bf{OR}}\) gates, \({\bf{AND}}\) gates, and inverters.

Short answer

Thus, it built\({\bf{x\bar y + \bar xy}}\) circuit.

Step-by-step solution

Definition

The complement of an element: \({\bf{\bar 0 = 1}}\) and \({\bf{\bar 1 = 0}}\).

The Boolean sum \({\bf{ + }}\) or \({\bf{OR}}\) is \({\bf{1}}\) if either term is \({\bf{1}}{\bf{.}}\)

The Boolean product \({\bf{ \bullet }}\) or \({\bf{AND}}\) is \({\bf{1}}\) if both terms are \({\bf{1}}{\bf{.}}\)

An inverter (Not gate) takes the complement of the input.

An \({\bf{AND}}\) gate takes the Boolean product of the input.

An \({\bf{OR}}\) gate takes the Boolean sum of the input.

Circuit

Let the two switches be \({\bf{x}}\) and \({\bf{y}}\). Let us assume the two switches are initially in the same position, when the light is off.

If exactly two switches are switched, then the light needs to turn on.

Therefore, the circuit below represents such a circuit, which note corresponds with the expression \({\bf{x\bar y + \bar xy}}\).