Boolean Algebra & Logic Prepared by Dr P Marais (Modified by D Burford)

Boolean Algebra & Logic Modern computing devices are digital –Use two states to represent all entities: 1 and 0 –Call these two logical states TRUE and FALSE All operations are on such values, and can only result in these values

Boolean Algebra & Logic George Boole formalised such a logic algebra: “Boolean Algebra” Modern digital circuits are designed and optimised using this theory We implement “functions” (such as add, compare etc) in hardware, using corresponding Boolean expressions

Boolean Operators There are 3 basic logic operators A, B are variables that can be TRUE or FALSE TRUE represented by 1; FALSE by 0 OperatorUsageNotation ANDA AND BA.B ORA OR BA+B NOTNOT A A

Truth Table: AND, OR, NOT To show the value of each operator we use a Truth Table –AND: only if both are TRUE –OR: if either is TRUE –NOT:inverts value ABF=A.BF=A+BF = AF=B

NAND, NOR and XOR –NAND: if either are FALSE [NOT (A AND B)] –NOR:if both are FALSE [NOT (A OR B)] –XOR:if either is TRUE, but not both ABF=A.BF=A+B F=A  B

Logic Gates These operators have symbolic representations: “logic gates” Building blocks for computer circuit desgin

Logic Gates

Finding a Boolean Representation F = F(A,B,C); F called “output variable” Find F values which are TRUE: –If A=0, B=1, C=0, then F = 1. –F 1 =A.B.C –F 2 = –F 3 = –F = ABCF

Finding a Boolean Representation F = F(A,B,C); F called “output variable” Find F values which are TRUE: –If A=0, B=1, C=0, then F = 1. –F 1 =A.B.C –F 2 = A.B.C –F 3 = A.B.C –F = F 1 + F 2 + F 3 ABCF

Algebraic Identities Commutative: A.B = B.A and A+B = B+A Distributive: A.(B+C) = (A.B) + (A.C) A+(B.C) = (A+B).(A+C) Associative: A.(B.C) = (A.B).C and A+(B+C) = (A+B)+C

Algebraic Identities Identitiy Elements: 1.A = A and 0 + A = A Inverse: A.A = 0 and A + A = 1 DeMorgan's Laws: A.B = A + B and A+B = A.B