Logic diagrams

Ben installs burglar alarms. The alarm is made up of a door sensor and a motion sensor. When the alarm is set, the siren will sound if either the door sensor or the motion sensor detect movement.

The alarm also has a test mode setting. When the test mode setting is enabled, an engineer can check the sensors are working without the siren going off.

The inputs to the alarm are as follows.

Draw a logic circuit to show the logic that is used in the burglar alarm to determine if the siren goes off.

Simplify the Boolean expression ¬A v ¬B using De Morgan’s First Law.

Simplify the Boolean expression ¬(¬B) using double negation.

Simplify the Boolean expression (A v B) ˄ (A v C) using distribution.

A computer scientist has created the following logic circuit shown in Fig. 6.

Give the Boolean expression that represents the logic circuit shown in Fig. 6. Do not attempt to simplify the expression.

Complete the truth table for the logic circuit shown in Fig. 6.

The following Karnaugh map represents another logic circuit.

Use this Karnaugh map to find the simplified expression for this circuit.

You should highlight the map as appropriate and write the expression here.

All users of a computer system have a unique username and password. The computer system has implemented two-factor authentication so that users must respond to either an email or text message containing a secret code to be able to access the system.

Let:

A be a Boolean value for if a user enters a valid username

B be a Boolean value for if a user enters a password that matches their username

C be a Boolean value for if a user is able to respond to an email containing a secret code

D be a Boolean value for if a user is able to respond to a text message containing a secret code

Q be a Boolean value for if entry to the computer system is allowed

Complete the Boolean expression below:

Another Boolean expression for a logic system is shown below:

Q ≡ ¬ (¬ A ¬ B)

Simplify this Boolean expression so that it does not include any negation. You must explain which Boolean algebra rule(s) you are using at each step.

The logic circuit above has two inputs (A, B) and two outputs (S, C).

Give the Boolean expressions for the outputs S and C.

Complete the truth table for this logic circuit.

Describe how this logic circuit can be adapted to add together two 4-bit binary numbers.

A company releases an in-home virtual assistant called ‘Bertie Butler’.

The device, when placed in a room, listens out for the phrase “Hey Bertie”. When someone says that phrase it then listens to the question that follows and tries to give a relevant answer.

Bertie Butler has a number of built-in input and output devices.

Name one input device and one output device that might be part of Bertie Butler. For each device give a reason for it being built into the virtual assistant.

Input Device Name:........

Input Device Reason:........

Output Device Name:........

Output Device Reason:........

The Bertie Butler device runs off an embedded operating system.

Define the term ‘embedded operating system’.

Bertie Butler’s circuitry is designed to only listen out for “Hey Bertie” under certain circumstances, which are:

The privacy button (P) must be off and the microphone must generate a signal (S) to say a sound has been heard.

Complete the truth table for whether the device is listening (L).

Draw logic gates to represent the circuitry needed.

The Bertie Butler machine uses a multicore processor.

Define the term ‘multicore processor’.

Draw a logic gate diagram to represent the Boolean expression

Q ≡ ¬A∨B

A half adder has the truth table shown below:

Draw a half adder using logic gates.

Draw the logic gates represented by the Karnaugh Map below. Show your working.