Boolean Algebra

Boolean algebra is basis of logic minimization.

• identities:

X + 0 = X X • 1 = X

X + 1 = 1 X • 0 = 0

• idempotence law:

X + X = X X • X = X

• complements law:

X + X´ = 1 X • X´ = 0

• commutative law:

(Latin commutativus - varying, exposed to transition) –

the law of logic theory on which, on the analogies of algebra, the outcome of the operation effected above two expressions, does not depend on order of these expressions. As in logic theory it is possible to multiply expressions (in conjunction) and to add (in the disjunction), that, according to the law of a commutability the outcome of addition (multiplying) does not depend on the order of summands (multipliers) and, therefore, operation of addition (and also multiplyings), i.e. conjunction and the disjunction of expressions, is commutative.

X + Y = Y + X X • Y = Y • X

• associative law:

(X + Y) + Z= X + (Y + Z)= X + Y + Z

(X • Y) • Z = X • (Y • Z) = X • Y • Z

• distributive law:

X • (Y+Z) = (X•Y) + (X•Z)

X + (Y•Z) = (X+Y) • (X+Z)

Theorems of Boolean Algebra

Proving Distributive and Covering Theorems

(X+Y) • (X+Z) = X + (Y•Z)

Proving Covering Theorem

X + X•Z = X

Duality – every boolean function has a dual

– algorithm:

1. change ANDs to ORs and vice versa (every AND change to OR and conversely);

2. change logic 0’s to logic 1’s and vice versa (every logic 0 change to 1 and conversely);

3. leave literals unchanged (every literal is unchanged).

{f(X₁,X₂,…,Xn,0,1,+,•)}^D = f(X₁,X₂,…,Xn,1,0,•,+)

If expression is true, then dual is true;

Duality theorem ≠ DeMorgan’s theorem

DeMorgan’s theorem: procedure for complementing a complex function.

(1) (X + Y + Z +…) = X •Y • Z …

(2) X •Y • Z … = X +Y + Z +…

– algorithm:

1. change ANDs to ORs and vice versa (every AND change to OR and conversely);

2. change logic 0’s to logic 1’s and vice versa (every logic 0 change to 1 and conversely);

3. complement each literal 9 every literal change to its negation).

– general rule: {f(X₁,X₂,…,Xn,0,1,+,•)}′ = f(X₁′,X₂′,…,Xn′,1,0,•,+)

Example:

(AVB)’óA’^B’ , (A^B)’óA’VB’

Example of DeMorgan’s

• Let’s use DeMorgan’s to simplify the following expression:

• F = (X • Y) + (A • B)

• compliments and duals

• Quick test

• Build a logic function that outputs a 1 if the input variables match the pattern:

• A=1 B=1 C=0

• a) A•B•C′

• b) A+B+C′

• c) A•B+C′

Recommended Books

• John Truss, “Discrete Mathematics for Computer Scientists”, Addison-Wesley, Second Edition, 1999

• Nimal Nissanke, “Introductory Logic and Sets for Computer Scientists”, Addison-Wesley, 1999

• James A Anderson, “Discrete Mathematics with Combinatorics (2nd edition)”, Prentice-Hall, 2004

• Rod Haggarty “Discrete mathematics for computing” Addison Wesley

• A Chetwynd and P Diggle, “Discrete Mathematics” Butterworth-Heinemann

• Geoffrey Finch, “How to study linguistics”, Macmillan, 1998

• J N Crossley and others, “What is mathematical logic?”, Oxford University Press, 1972