INTRO. TO COMP. ENG. •CHAPTER III CHAPTER III-1 BOOLEAN ALGEBRA CHAPTER III BOOLEAN ALGEBRA R.M. Dansereau; v.1.0
BOOLEAN VALUES INTRO. TO COMP. ENG. •BOOLEAN VALUES CHAPTER III-2 INTRODUCTION BOOLEAN ALGEBRA • Boolean algebra is a form of algebra that deals with single digit binary values and variables. • Values and variables can indicate some of the following binary pairs of values: • ON / OFF • TRUE / FALSE • HIGH / LOW • CLOSED / OPEN • 1 / 0 R.M. Dansereau; v.1.0
BOOL. OPERATIONS INTRO. TO COMP. ENG. •BOOLEAN VALUES -INTRODUCTION CHAPTER III-3 FUNDAMENTAL OPERATORS BOOLEAN ALGEBRA • Three fundamental operators in Boolean algebra A A ′ ∼ • NOT : unary operator that complements represented as , , or A • AND : binary operator which performs logical multiplication ⋅ • i.e. A ANDed with B would be represented as AB or A B • OR : binary operator which performs logical addition • i.e. ORed with would be represented as A B A B + NOT AND OR A B AB A B A B + A A 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 1 1 1 R.M. Dansereau; v.1.0
BOOL. OPERATIONS INTRO. TO COMP. ENG. •BOOLEAN OPERATIONS -FUNDAMENTAL OPER. CHAPTER III-4 BINARY BOOLEAN OPERATORS BOOLEAN ALGEBRA • Below is a table showing all possible Boolean functions F N given the two- inputs A and B . F 0 F 1 F 2 F 3 F 4 F 5 F 6 F 7 F 8 F 9 F 10 F 11 F 12 F 13 F 14 F 15 A B 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 AB A B A B B A AB 1 + Null Identity ⊕ ⊕ A B A B Inhibition Implication A B + R.M. Dansereau; v.1.0
BOOLEAN ALGEBRA INTRO. TO COMP. ENG. •BOOLEAN OPERATIONS -FUNDAMENTAL OPER. CHAPTER III-5 -BINARY BOOLEAN OPER. PRECEDENCE OF OPERATORS BOOLEAN ALGEBRA • Boolean expressions must be evaluated with the following order of operator precedence • parentheses • NOT Example: • AND ( ( ) ) E F A C BD BC = + + • OR F A C B D B C E = + + { { { R.M. Dansereau; v.1.0
BOOLEAN ALGEBRA INTRO. TO COMP. ENG. •BOOLEAN OPERATIONS •BOOLEAN ALGEBRA CHAPTER III-6 -PRECEDENCE OF OPER. FUNCTION EVALUATION BOOLEAN ALGEBRA • Example 1: Evaluate the following expression when A 1 , B 0 , C 1 = = = F C CB BA = + + • Solution ⋅ ⋅ F 1 1 0 0 1 1 0 0 1 = + + = + + = • Example 2: Evaluate the following expression when A 0 , B 0 , C 1 , D 1 = = = = ( ( ) ) F D BCA AB C C = + + + • Solution ⋅ ( ⋅ ⋅ ( ⋅ ) ) ⋅ ( ) ⋅ F 1 0 1 0 0 0 1 1 1 0 1 1 1 1 1 = + + + = + + = = R.M. Dansereau; v.1.0
BOOLEAN ALGEBRA INTRO. TO COMP. ENG. •BOOLEAN OPERATIONS •BOOLEAN ALGEBRA CHAPTER III-7 -PRECEDENCE OF OPER. BASIC IDENTITIES BOOLEAN ALGEBRA -FUNCTION EVALUATION ⋅ X 0 X X 1 X + = = Identity ⋅ X 1 1 X 0 0 + = = ⋅ X X X X X X + = = Idempotent Law X ′ X X ′ ⋅ X 1 0 + = = Complement ( X ′ )′ X = Involution Law X Y Y X XY YX + = + = Commutativity ( ) ( ) ( ) ( ) Z X Y Z X Y Z X YZ XY + + = + + = Associativity ( ) ( ) X ( ) X Y Z XY XZ X YZ X Y Z + = + + = + + Distributivity ( ) X XY X X X Y X + = + = Absorption Law X ′ Y X X ′ ( ) X X Y Y XY + = + + = Simplification ( )′ X ′ Y ′ ( )′ X ′ Y ′ X Y XY + = = + DeMorgan’s Law X ′ Z ( ) X ′ ( ) Y ( ) XY YZ X Y Z Z + + + + + Consensus Theorem X ′ Z ( ) X ′ ( ) XY = + X Y Z = + + R.M. Dansereau; v.1.0
BOOLEAN ALGEBRA INTRO. TO COMP. ENG. •BOOLEAN ALGEBRA -PRECEDENCE OF OPER. CHAPTER III-8 -FUNCTION EVALUATION DUALITY PRINCIPLE BOOLEAN ALGEBRA -BASIC IDENTITIES • Duality principle: • States that a Boolean equation remains valid if we take the dual of the expressions on both sides of the equals sign. • The dual can be found by interchanging the AND and OR operators along with also interchanging the 0 ’s and 1 ’s. • This is evident with the duals in the basic identities. • For instance: DeMorgan’s Law can be expressed in two forms ( )′ X ′ Y ′ ( )′ X ′ Y ′ as well as X Y XY + = = + R.M. Dansereau; v.1.0
BOOLEAN ALGEBRA INTRO. TO COMP. ENG. •BOOLEAN ALGEBRA -FUNCTION EVALUATION CHAPTER III-9 -BASIC IDENTITIES FUNCTION MANIPULATION (1) BOOLEAN ALGEBRA -DUALITY PRINCIPLE • Example: Simplify the following expression F BC BC BA = + + • Simplification ( ) F B C C BA = + + ⋅ F B 1 BA = + ( ) F B 1 A = + F B = R.M. Dansereau; v.1.0
BOOLEAN ALGEBRA INTRO. TO COMP. ENG. •BOOLEAN ALGEBRA -BASIC IDENTITIES CHAPTER III-10 -DUALITY PRINCIPLE FUNCTION MANIPULATION (2) BOOLEAN ALGEBRA -FUNC. MANIPULATION • Example: Simplify the following expression F A AB ABC ABCD ABCDE = + + + + • Simplification ( ) F A A B BC BCD BCDE = + + + + F A B BC BCD BCDE = + + + + ( ) F A B B C CD CDE = + + + + F A B C CD CDE = + + + + ( ) F A B C C D DE = + + + + F A B C D DE = + + + + F A B C D E = + + + + R.M. Dansereau; v.1.0
BOOLEAN ALGEBRA INTRO. TO COMP. ENG. •BOOLEAN ALGEBRA -BASIC IDENTITIES CHAPTER III-11 -DUALITY PRINCIPLE FUNCTION MANIPULATION (3) BOOLEAN ALGEBRA -FUNC. MANIPULATION • Example: Show that the following equality holds ( ) ( ) B ( ) A BC BC A B C C + = + + + • Simplification ( ) ( ) A BC BC A BC BC + = + + ( ) BC ( ) A BC = + ( ) B ( ) A B C C = + + + R.M. Dansereau; v.1.0
STANDARD FORMS INTRO. TO COMP. ENG. •BOOLEAN ALGEBRA -BASIC IDENTITIES CHAPTER III-12 -DUALITY PRINCIPLE SOP AND POS BOOLEAN ALGEBRA -FUNC. MANIPULATION • Boolean expressions can be manipulated into many forms. • Some standardized forms are required for Boolean expressions to simplify communication of the expressions. • Sum-of-products ( SOP ) • Example: ( , , , ) F A B C D AB BCD AD = + + • Products-of-sums ( POS ) • Example: ( , , , ) ( ) B ( ) A ( ) F A B C D A B C D D = + + + + R.M. Dansereau; v.1.0
STANDARD FORMS INTRO. TO COMP. ENG. •BOOLEAN ALGEBRA •STANDARD FORMS CHAPTER III-13 -SOP AND POS MINTERMS BOOLEAN ALGEBRA • The following table gives the minterms for a three-input system m 0 m 1 m 2 m 3 m 4 m 5 m 6 m 7 A B C ABC ABC ABC ABC ABC ABC ABC ABC 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 1 R.M. Dansereau; v.1.0
STANDARD FORMS INTRO. TO COMP. ENG. •BOOLEAN ALGEBRA •STANDARD FORMS CHAPTER III-14 -SOP AND POS SUM OF MINTERMS BOOLEAN ALGEBRA -MINTERMS • Sum-of-minterms standard form expresses the Boolean or switching expression in the form of a sum of products using minterms . • For instance, the following Boolean expression using minterms ( , , ) F A B C ABC ABC ABC ABC = + + + could instead be expressed as ( , , ) F A B C m 0 m 1 m 4 m 5 = + + + or more compactly ∑ ( , , ) ( , , , ) ( , , , ) F A B C m 0 1 4 5 one-set 0 1 4 5 = = R.M. Dansereau; v.1.0 Minterms are products, so� this is called a� "sum of products" (SOP)
STANDARD FORMS INTRO. TO COMP. ENG. •STANDARD FORMS -SOP AND POS CHAPTER III-15 -MINTERMS MAXTERMS BOOLEAN ALGEBRA -SUM OF MINTERMS • The following table gives the maxterms for a three-input system M 0 M 1 M 2 M 3 M 4 M 5 M 6 M 7 A B C A B C A B C A B C + + + + + + + + A B C A B C A B C A B C + + + + + + + + A B C 0 0 0 0 1 1 1 1 1 1 1 0 0 1 1 0 1 1 1 1 1 1 0 1 0 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 R.M. Dansereau; v.1.0
STANDARD FORMS INTRO. TO COMP. ENG. •STANDARD FORMS -MINTERMS CHAPTER III-16 -SUM OF MINTERMS PRODUCT OF MAXTERMS BOOLEAN ALGEBRA -MAXTERMS • Product-of-maxterms standard form expresses the Boolean or switching expression in the form of product of sums using maxterms . • For instance, the following Boolean expression using maxterms ( , , ) ( ) A ( ) A ( ) F A B C A B C B C B C = + + + + + + could instead be expressed as ( , , ) ⋅ ⋅ F A B C M 1 M 4 M 7 = or more compactly as ∏ ( , , ) ( , , ) ( , , ) F A B C M 1 4 7 zero-set 1 4 7 = = R.M. Dansereau; v.1.0
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