1.4.3 Boolean Algebra

Exam-Style Questions (37 marks)

1. Complete the truth table for XOR

   A	B	Q
   0	0	0
   0	1	1
   1	0	1
   1	1	0

2. The truth table of a NAND gate is shown below.

   A	B	Q
   0	0	1
   0	1	1
   1	0	1
   1	1	0

   Construct a sequence of gates equivalent to a NAND gate, but built only of AND, OR, and NOT gates.

   ¬(A ∧ B)

          ┌──────╮
   A ─────┤      │  │╲
          │      ├──┤ ├◯───── Q
   B ─────┤      │  │╱
          └──────╯
   

3. Create a truth table to represent the following Boolean expression: Q = ¬(A ∧ B) ∨ C

   A	B	C	Q
   0	0	0	1
   0	0	1	1
   0	1	0	1
   0	1	1	1
   1	0	0	1
   1	0	1	1
   1	1	0	0
   1	1	1	1

4. 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:

   Q = (A ∧ B) ∨ (C ∧ D)

5. Burger House is a fast food restaurant which wants to encourage healthy eating amongst its younger diners.

   Shown below is the Burger House children's menu.

   Children's Menu

   Burgers

   Cheeseburger
   Grilled Chicken burger (Healthy Option)

   ---

   Side Dishes

   French Fries
   Salad (Healthy Option)
   Carrot Sticks (Healthy Option)

   ---

   Desserts

   Chocolate Brownie
   Fruit Salad (Healthy Option)

   Children receive a free toy when they select a meal (i.e. one burger, one side dish and one dessert) made up of only healthy options.

   • Let g be a Boolean value for if a child has chosen a grilled chicken burger.
   • Let s be a Boolean value for if a child has chosen salad.
   • Let c be a Boolean value for if a child has chosen carrot sticks.
   • Let f be a Boolean value for if a child has chosen fruit salad.
   • Let t be a Boolean value for whether a child receives a toy.

   1. Write an expression using Boolean algebra to determine whether a child receives a toy when they select a meal.

      t = g ∧ (s ∨ c) ∧ f

   2. Burger House wants to add this logic into its till system. Complete the code below, assuming that g, s, c, f, and t are Boolean variables with the same meaning as in part (a).

      t = false
      if g and (s or c) and f then
          t = true
      endif
      

6. An electronics engineer needs a circuit with the following logic:

   (A ∧ B) ∨ (¬A ∧ B) ∨ (¬C ∧ ¬D)

   Complete and use the Karnaugh map below to simplify the expression above.

   ​ ╲AB CD╲ ​	00	01	11	10
   00	1	1	1	1
   01	0	1	1	0
   11	0	1	1	0
   10	0	1	1	0

   Simplified expression: B ∨ ¬(C ∨ D)

7. ​ ╲AB CD╲ ​	00	01	11	10
   00	1	1	0	1
   01	0	0	0	0
   11	0	0	1	0
   10	1	1	1	0

   State the Boolean expression represented by the Karnaugh map above, in its smallest form

   (¬A ∧ ¬D) ∨ (¬B ∧ ¬C ∧ ¬D) ∨ (A ∧ B ∧ C)