The document discusses Boolean algebra and its basic concepts. Boolean algebra uses 1 to represent True and 0 to represent False. It defines basic Boolean operations like AND, OR, and NOT. The document also shows an example of a Boolean equation to represent an "umbrella" proposition based on weather conditions. It then works through reducing a Boolean function step-by-step using Boolean identities and laws like commutative, associative, and distributive properties.

1. BO
O
LEAN
ALG
EBRA

2. 2
Boolean Algebra
I will take an umbrella with me if it is raining
or the weather forecast is bad

3. 3
Boolean Algebra
I will take an umbrella with me if it is raining
or the weather forecast is bad

4. 4
Boolean Algebra
I will take an umbrella with me if it is raining
or the weather forecast is bad
A Simple Proposition

5. 5
Boolean Algebra
I will take an umbrella with me if it is raining
or the weather forecast is bad
Propositions may be TRUE or FALSE, are functions of other
propositions, and connected by logical connections (AND, OR,
NOT)

6. 6
Boolean Algebra
I will take an umbrella with me if it is raining
or the weather forecast is bad
Truth Table

7. 7
Boolean Algebra
George Boole, the mathematician, developed
boolean algebra to simplify the handling of
complex connectives.
Boolean Algebra uses ordinary algebraic
notation, and 1 for True and 0 for False.

8. 8
Boolean Algebra
I will take an umbrella with me if it is raining
or the weather forecast is bad
U = R + F
Boolean equation
Raining Bad Forecast Umbrella
0 0 0
0 1 1
1 0 1
1 1 1

9. 9
Boolean Algebra
If I do not take the car then I will take an
umbrella if it is raining or the weather forecast
is bad

10. 10
Boolean Algebra
If I do not take the car then I will take an
umbrella if it is raining or the weather forecast
is bad

11. 11
Boolean Algebra
If I do not take the car then I will take an
umbrella if it is raining or the weather forecast
is bad

12. 12
Boolean Algebra
If I do not take the car then I will take an
umbrella if it is raining or the weather forecast
is bad
U = C’. (W + R)

13. 13
Boolean Algebra
U = C’. (W + R)
If I do not take the car then I will take an
umbrella if it is raining or the weather forecast
is bad

14. 14
Boolean Identities
Basic boolean identities.
X + 0 = X X * 1 = X X + 1 = 1
X * 0 = 0 X + X = X X * X = X
X + X = 1 X = X X * X = 0

15. 15
Basic Laws of Boolean Algebra
The basic laws of boolean algebra are:
Commutative Law:
X + Y = Y + X XY = YX
Associative Law:
X+(Y+Z) = (X+Y)+Z X(YZ)=(XY)Z
Distributive Law:
X(Y+Z) = XY + XZ X+YZ = (X+Y)(X+Z)

16. 16
Boolean Laws and Identities
Reduce the boolean function F = X Y Z + X Y + X Y Z

17. 17
Boolean Laws and Identities
Reduce the boolean function F = X Y Z + X Y + X Y Z
F = X Y (Z + Z) + X Y (Distributive and Commutative Laws)

18. 18
Boolean Laws and Identities
Reduce the boolean function F = X Y Z + X Y + X Y Z
F = X Y (Z + Z) + X Y (Distributive and Commutative Laws)
F = X Y (Z + Z) + X Y (Identity)
= 1

19. 19
Boolean Laws and Identities
Reduce the boolean function F = X Y Z + X Y + X Y Z
F = X Y (Z + Z) + X Y (Distributive and Commutative Laws)
F = X Y (Z + Z) + X Y (Identity)
= 1
F = X Y + X Y

20. 20
Boolean Laws and Identities
Reduce the boolean function F = X Y Z + X Y + X Y Z
F = X Y (Z + Z) + X Y (Distributive and Commutative Laws)
F = X Y (Z + Z) + X Y (Identity)
= 1
F = X Y + X Y
F = (X + X) Y (Distributive Law)

21. 21
Boolean Laws and Identities
Reduce the boolean function F = X Y Z + X Y + X Y Z
F = X Y (Z + Z) + X Y (Distributive and Commutative Laws)
F = X Y (Z + Z) + X Y (Identity)
= 1
F = X Y + X Y
F = (X + X) Y (Distributive Law)
F = (X + X) Y (Identity)
= 1

22. 22
Boolean Laws and Identities
Reduce the boolean function F = X Y Z + X Y + X Y Z
F = X Y (Z + Z) + X Y (Distributive and Commutative Laws)
F = X Y (Z + Z) + X Y (Identity)
= 1
F = X Y + X Y
F = (X + X) Y (Distributive Law)
F = (X + X) Y (Identity)
= 1
F = Y

23. 23
Advanced Boolean Laws
Here are some advanced laws of boolean algebra that
can be directly applied to the reduction of boolean
functions.
X + X Y = X
X Y + X Y = X
X + X Y = X + Y
X (X + Y) = X
(X + Y)(X + Y) = X
X (X + Y) = X