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

2. 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

3. 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

4. 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

5. 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 000011 010110 100101 111100

6. 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 00110 01101 10101 11000

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

8. Logic Gates

9. 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 0000 0010 0101 0111 1000 1010 1101 1110

10. 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 0000 0010 0101 0111 1000 1010 1101 1110

11. 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

12. 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