# CMSC 313 Lecture 17 Postulates & Theorems of Boolean Algebra Semiconductors CMOS Logic Gates

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1 CMSC 313 Lecture 17 Postulates & Theorems of Boolean Algebra Semiconductors CMOS Logic Gates UMBC, CMSC313, Richard Chang

2 Last Time Overview of second half of this course Logic gates & symbols Equivalence of Boolean functions, truth tables, Boolean formulas and combinational circuits Universality of NAND gates UMBC, CMSC313, Richard Chang

3 Postulates of Boolean Algebra Commutative: AB = BA, A + B = B + A Associative: (AB)C = A(BC), (A + B) + C = A + (B + C) Distributive: A(B + C) = AB + AC, A + BC = (A + B)(A + C) Identity: there exists 0 and 1 such that for all A, 1A = A and 0 + A = A. Complement: for all A, there exists A such that AA = 0 and A + A = 1 where 0 and 1 are the identity elements. 1

4 Some Theorems of Boolean Algebra Zero and One: 0A = 0, 1 + A = 1 Idempotence: AA = A, A + A = A Involution: A = A DeMorgan s: AB = A + B, A + B = AB Absorption: A(A + B) = A, A + AB = A Consensus: AB + AC + BC = AB + AC, (A + B)(A + C)(B + C) = (A + B)(A + C) 2

5 Theorems are derived from the postulates Idempotence: A = A1 identity, commutative = A(A + A) complement = AA + AA distributive = AA + 0 complement = AA commutative, identity A = 0 + A identity = (AA) + A complement = (A + A)(A + A) distributive = (A + A)1 commutative, complement = A + A commutative, identity 3

6 Theorems are derived from the postulates Zero and One: 0A = (AA)A complement = A(AA) commutative = (AA)A associative = AA idempotent = 0 complement 1 + A = (A + A) + A complement = A + (A + A) commutative = (A + A) + A associative = A + A idempotent = 1 complement 4

7 Theorems are derived from the postulates Absorption: A + AB = A1 + AB identity, commutative = A(1 + B) distributive = A1 one = A commutative, identity A(A + B) = AA + AB distributive = A + AB idempotent = A absorption 5

8 Proof by truth table? Absorption: A + AB = A A B AB A + AB Proof by truth table only applies to 0-1 Boolean Algebra. Proof by derivation from postulates holds for any Boolean Algebra. 6

9 A Boolean Algebra with 8 elements Elements are the subsets of {a, b, c}:, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} Operations: AB A B, A + B A B. Union and intersection are commutative and associative. Union distributes over intersection: A (B C) = (A B) (A C). Intersection distributes over union: A (B C) = (A B) (A C). Identity: 0 =, 1 = {a, b, c}, {a, b, c} A = A and A = A. Complement: A = {a, b, c} A, A A = and A A = {a, b, c}. All postulates hold. Therefore, all derived theorems also hold. 7

10 Simplifying MAJ3 Simplify MAJ3 using the postulates and theorems of Boolean Algebra MAJ3(A, B, C) = ABC + ABC + ABC + ABC SOP form = ABC + ABC + ABC + ABC + ABC + ABC idempotent = ABC + ABC + ABC + ABC + ABC + ABC commutative = (A + A)BC + (B + B)AC + (C + C)AB distributive = 1BC + 1AC + 1AB complement = BC + AC + AB identity Resulting circuit uses fewer gates. 8

11 B-6 Appendix B: Reduction of Digital Logic The Algebraic Method This majority circuit is functionally equivalent to the previous majority circuit, but this one is in its minimal two-level form: A B C F Principles of Computer Architecture by M. Murdocca and V. Heuring 1999 M. Murdocca and V. Heuring

12 A-19 Appendix A: Digital Logic AND-OR Implementation of Majority A B C Gate count is 8, gate input count is 19. A B C A B C F A B C A B C Principles of Computer Architecture by M. Murdocca and V. Heuring 1999 M. Murdocca and V. Heuring

13 How do we make gates??? UMBC, CMSC313, Richard Chang

14 A-5 Appendix A: Digital Logic A Truth Table Developed in 1854 by George Boole. Further developed by Claude Shannon (Bell Labs). Outputs are computed for all possible input combinations (how many input combinations are there?) Consider a room with two light switches. How must they work? GND Inputs Output Hot Light Z A B Z Switch A Switch B Principles of Computer Architecture by M. Murdocca and V. Heuring 1999 M. Murdocca and V. Heuring

15 Electrically Operated Switch Example: a relay source: UMBC, CMSC313, Richard Chang

16 Semiconductors Electrical properties of silicon Doping: adding impurities to silicon Diodes and the P-N junction Field-effect transistors UMBC, CMSC313, Richard Chang

17 Los Alamos National Laboratory's Chemistry Division Presents Periodic Table of the Elements Period Group 1 IA 1A 18 IIIA 8A V 1 1 H IIA 2A 13 IIIA 3A 14 IVA 4A 15 VA 5A 16 VIA 6A 17 VIIA 7A 2 He Li Be B C N O F Ne Na Mg IIIB 3B 4 IVB 4B 5 VB 5B 6 VIB 6B 7 VIIB 7B IB 1B 12 IIB 2B 13 Al Si P S Cl Ar VIII K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc (98) 44 Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cs Ba * La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po (210) 85 At (210) 86 Rn (222) 7 87 Fr (223) 88 Ra (226) 89 ~ Ac (227) 104 Rf (257) 105 Db (260) 106 Sg (263) 107 Bh (262) 108 Hs (265) 109 Mt (266) () () () () () () Lanthanide Series* 58 Ce Pr Nd Pm (147) 62 Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Actinide Series~ 90 Th Pa (231) 92 U (238) 93 Np (237) 94 Pu (242) 95 Am (243) 96 Cm (247) 97 Bk (247) 98 Cf (249) 99 Es (254) 100 Fm (253) 101 Md (256) 102 No (254) 103 Lr (257)

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37 An Inverter using MOSFET CMOS = complementary metal oxide semiconductor P-type transistor conducts when gate is low N-type transistor conducts when gate is high +5v p-type MOSFET +5v +5v A z A z A z n-type MOSFET GND GND GND UMBC, CMSC313, Richard Chang

38 +5v +5v NAND GATE A B z A z A z B B GND +5v GND +5v +5v A z A z A z B B B GND GND GND

39 +5v +5v NOR GATE A B z A A B z B z GND GND +5v +5v +5v A A A B z B z B z GND GND GND

40 CMOS Logic vs Bipolar Logic MOSFET transistors are easier to miniaturize CMOS logic has lower current drain CMOS logic is easier to manufacture UMBC, CMSC313, Richard Chang

41 Circuits for Addition Next time UMBC, CMSC313, Richard Chang

42 References Materials on semiconductors, PN junction and transistors taken from the HyperPhysics web site: < UMBC, CMSC313, Richard Chang

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