Boolean variable ◦ 0 or 1

Boolean Algebra ◦ Entities 0 and 1 together with the operators AND,

OR, NOT

B. A. = {0, 1 | AND, OR, NOT}

X NOT (X)/ X

0 1

1 0

Truth Table

X Y X AND Y/ (X.Y)

0 0 0

0 1 0

1 0 0

1 1 1

X Y X OR Y/ (X+Y)

0 0 0

0 1 1

1 0 1

1 1 1

1. X + 0 = X

2. X + 1 = 1

3. X . 0 = 0

4. X . 1 = X

5. X + X = X

6. X . X = X

7. X + X = 1

8. X . X = 0

9. (X) = X

10. Commutative Law

X+Y=Y+X

X.Y=Y.X

Perfect induction employs a truth table which describes the validity of the Boolean entity for all the possible value combinations of the Boolean variables.

I will buy a house if I get a salary increase or if I win the lottery.

Salary Increase Win Lottery

Buy a house = Salary

Increase or Win

Lottery

False False False

False True True

True False True

True True True

X = "get a salary increase“ Y = "win the lottery" F = "buy a house“

X Y F = X + Y

0 0 0

0 1 1

1 0 1

1 1 1

1. I will eat lunch if I am hungry and if lunch is ready.

2. I will buy a new book if the old book is finished or if it is a new subject.

1. X + 0 = X

X X + 0

0 0

1 1

1. X + X.Y = X

2. X. (X. Y) = X

3. X + X.Y = X+Y

Logic gate

It is a digital circuit that performs common logical functions such AND, OR, NOT.

NOT gate ( Inverter) ◦ Performs logical complementation.

X NOT (X)/ X

0 1

1 0

X Z

AND gate ◦ Performs logical multiplication

X

Y

Z

OR gate ◦ Performs logical addition

X

Y

Z

Z = X + X.Y

X

Y

X.Y

X

Z = X + X.Y

1. Z = X. (X + Y)

2. Z = X + X.Y

Why is it better to simplify the Boolean expression?

Z = [(A + B) (A + C) ] Draw logic gate circuit

= A.A + A.C + A.B + C.B

= A+ AC + AB + CB

= A ( 1 + C + B) + CB

= A + CB Draw logic gate circuit

1. Z = A.B + A.B.C + A.B.C.D + A.B.C.D.E

2. Z = ABC [ AB + C (BC + AC)]