1. C.S. Choy21 BOOLEAN ALGEGRA The Mathematics of logic Boolean variables have only two possible values (binary) Operators:. Product+ SumComplement A.B A+B A

2. C.S. Choy22 BOOLEAN ALGEBRA Properties Associative – (A+B)+C=A+(B+C)=A+B+C (AB)C=A(BC)=ABC Commutative – A+B=B+A AB=BA Distributive – A(B+C)=AB+AC Others –

3. C.S. Choy23 BOOLEAN ALGEBRA Other Properties A+AB = A Proof: A+AB = A+B Proof:

4. C.S. Choy24 DeMORGAN’S THEOREM The complement of the SUM function is equal to the PRDUCT function of the complements A+B = AB Equivalent AB = A+B Expansion A+B+C = ABC ABC = A+B+C

5. C.S. Choy25 BOOLEAN ALGEBRA Expression Manipulation (A+B+C)(A+B+C) =

6. C.S. Choy26 TRUTH TABLE Tabulate all possible value combinations of an expression Proof of DeMorgan’s Theorem A+B = AB ABA+B 00110011 01010101 ABAB 00110011 01010101

7. C.S. Choy27 LOGIC GATES Building blocks of digital circuits AND Gate Output = AB ABoutput 00110011 01010101 00010001

8. C.S. Choy28 LOGIC GATES OR Gate Output = A + B ABoutput 00110011 01010101 01110111

9. C.S. Choy29 LOGIC GATES Inverter output = A Aoutput 01 10

10. C.S. Choy30 COMPLETE SET OF OPERATIONS OR, AND and INVERTER together form a complete set because any boolean function can be constructed from a combination of these three gates

11. C.S. Choy31 OTHER KINDS OF GATE NAND Itself a complete set NOR Itself a complete set

12. C.S. Choy32 OTHER KINDS OF GATE Exclusive-OR Gate This is useful as it is functionally equivalent to binary addition XOR = AB + AB = A + B Properties: –CommutativeA + B = B + A –Associative(A + B) + C = A + (B + C) –DistributiveA(B + C) = AB + AC ABA + B 00110011 01010101 01100110

13. C.S. Choy33 EXPRESSION OF DE-MORGAN’S THEOREM IN TERMS OF LOGIC GATES A + B = AB

14. C.S. Choy34 DESIGN PROCESS F = ABC ABcF 0000111100001111 0011001100110011 0101010101010101 0000000100000001 The term ABC can be written directly from the truth table as it corresponds with the binary pattern 111

15. C.S. Choy35 DESIGN PROCESS Example This is usually called a sum-of-products (SOP) configuration ABcF 0000111100001111 0011001100110011 0101010101010101 0010010000100100

16. C.S. Choy36 PRODUCT-OF-SUM (POS) CONFIGURATION ABcF 0000111100001111 0011001100110011 0101010101010101 0010010000100100

17. C.S. Choy37 DESIGN ALTERNATIVE USING BOOLEAN ALGBRA Fully NAND Implementation F = B + A(C + D)

18. C.S. Choy38 DESIGN ALTERNATIVE USING BOOLEAN ALGEBRA Fully NOR Implementation F = B + A(C + D) = B + AC + AD = B + A + C + A + D F = B + A(C + D) = B A (C + D) = B (A + C + D) =AB + B C+D = A + B + B + C + D