Combining Boolean operators

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.

Use the Karnaugh map to find a simplified Boolean expression that is equivalent to (¬ A ¬ B) (A ¬ B)

State the purpose of a D-type flip-flop circuit.

Describe the inputs and outputs used by a D-type flip-flop circuit, explaining how the inputs are used to control the outputs.

Draw a logic gate diagram to represent the Boolean expression

Q ≡ ¬A∨B

A cinema offers discounted tickets, but only under one of the following conditions:

• Customer is under 18 and has a student card.

• Customer is over 60 and has ID which proves this.

Let:

A be Customer is under 18

B be Customer has a student card

C be Customer is over 60

D be Customer has ID

Q be Discount ticket issued

Complete the Boolean expression below:

The cinema has a voucher which promises free popcorn when the voucher is produced whilst buying a soft drink or bottle of water.

Let:

E be Voucher is shown

F be Soft drink is bought

G be Bottle of water is bought

R be Free popcorn given.

This could be written as:

R ≡ (E ∧ F) ∨ (E ∧ G)

Complete the truth table below.

Simplify the expression

(E ∧ F) ∨ (E ∧ G)

Most films are now distributed to cinemas digitally. A studio allows cinemas to download its latest film 5 days before the release date via a private download. It wants to ensure that no cinema shows it before the release date.

Describe one technical measure the studio could use to ensure that films are not shown early.

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.