Laws of Boolean Algebra

• Commutative Law

• Associative Law

• Distributive Law

• Identity Law

• De Morgan's Theorem

Introduction• A set of rules formulated by the English mathematician George Boole

– Describe certain propositions whose outcome would be either true or false.

–AND represented with ( . ) A . B is usually written as AB

– OR represented with ( + )

– With regard to digital logic, these rules are used to describe circuits• State can be either, 1 (true) or 0 (false).

• P1: X = 0 or X = 1 • P2: 0 . 0 = 0 • P3: 1 + 1 = 1 • P4: 0 + 0 = 0 • P5: 1 . 1 = 1 • P6: 1 . 0 = 0 . 1 = 0 • P7: 1 + 0 = 0 + 1 = 1

Laws of Boolean Algebra

• Commutative Law • (a) A + B = B + A

(b) A B = B A

• Associate Law • (a) (A + B) + C = A + (B + C)

(b) (A B) C = A (B C)

• Distributive Law • (a) A (B + C) = A B + A C

(b) A + (B C) = (A + B) (A + C)

• Identity Law • (a) A + A = A

(b) A A = A

• Redundance Law • (a) A + A B = A

(b) A (A + B) = A

De Morgan’s Laws______ _ _

(A + B) = A B

____ _ _

(A B) = A + B

Other• (a) 0 + A = A

(b) 0 A = 0

• (a) 1 + A = 1 (b) 1 A = A

_• AB + AB = A _• (A + B)(A + B) = A _• A + A = 1 _• A A = 0 _• A + A B = A + B _• A ( A + B) = A B

_

Prove: A + A B = A + B

(1) Algebraically

(2) Using a truth table

Simplifying Expressions(1) Using the laws given simplify the following

expression:

_ _ _

Z = (A + B + C)(A + BC)

Example1. To get into a physics program in university, Samantha

needs to have OAC physics and either OAC algebra or OAC calculus. Assign Boolean variables to the conditions and write a Boolean expression for the program requirements. – Let P represent whether or not Samantha has OAC physics.

2. Another way of stating the conditions for the physics program is that Samantha needs OAC physics and OAC algebra, or OAC physics and OAC calculus. Using the same Boolean variables as above, write a Boolean expression for the program requirements.  

3. Since both of these expressions refer to the same situation the Boolean expressions must be equal. Verify this statement by comparing the truth tables for the expressions.