# CMSC 313 Lecture 16 Postulates & Theorems of Boolean Algebra Semiconductors CMOS Logic Gates

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1 CMSC 33 Lecture 6 Postulates & Theorems of oolean lgebra Semiconductors CMOS Logic Gates UMC, CMSC33, Richard Chang

2 Last Time Overview of second half of this course Logic gates & symbols Equivalence of oolean functions, truth tables, oolean formulas and combinational circuits Universality of NND gates UMC, CMSC33, Richard Chang

3 Postulates of oolean lgebra Commutative: =, + = + ssociative: ()C = (C), ( + ) + C = + ( + C) Distributive: ( + C) = + C, + C = ( + )( + C) Identity: there exists and such that for all, = and + =. Complement: for all, there exists such that = and + = where and are the identity elements.

4 Some Theorems of oolean lgebra Zero and One: =, + = Idempotence: =, + = Involution: = DeMorgan s: = +, + = bsorption: ( + ) =, + = Consensus: + C + C = + C, ( + )( + C)( + C) = ( + )( + C) 2

5 Theorems are derived from the postulates Idempotence: = identity, commutative = ( + ) complement = + distributive = + complement = commutative, identity = + identity = () + complement = ( + )( + ) distributive = ( + ) commutative, complement = + commutative, identity 3

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

7 Theorems are derived from the postulates bsorption: + = + identity, commutative = ( + ) distributive = one = commutative, identity ( + ) = + distributive = + idempotent = absorption 5

8 Proof by truth table? bsorption: + = + Proof by truth table only applies to - oolean lgebra. Proof by derivation from postulates holds for any oolean lgebra. 6

9 oolean lgebra with 8 elements Elements are the subsets of {a, b, c}:, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} Operations:, +. Union and intersection are commutative and associative. Union distributes over intersection: ( C) = ( ) ( C). Intersection distributes over union: ( C) = ( ) ( C). Identity: =, = {a, b, c}, {a, b, c} = and =. Complement: = {a, b, c}, = and = {a, b, c}. ll postulates hold. Therefore, all derived theorems also hold. 7

10 Simplifying MJ3 Simplify MJ3 using the postulates and theorems of oolean lgebra MJ3(,, C) = C + C + C + C SOP form = C + C + C + C + C + C idempotent = C + C + C + C + C + C commutative = ( + )C + ( + )C + (C + C) distributive = C + C + complement = C + C + identity Resulting circuit uses fewer gates. 8

11 -6 ppendix : Reduction of Digital Logic The lgebraic Method This majority circuit is functionally equivalent to the previous majority circuit, but this one is in its minimal two-level form: C F Principles of Computer rchitecture by M. Murdocca and V. Heuring 999 M. Murdocca and V. Heuring

12 -9 ppendix : Digital Logic ND-OR Implementation of Majority C Gate count is 8, gate input count is 9. C C F C C Principles of Computer rchitecture by M. Murdocca and V. Heuring 999 M. Murdocca and V. Heuring

13 How do we make gates??? UMC, CMSC33, Richard Chang

14 -5 ppendix : Digital Logic Truth Table Developed in 854 by George oole. Further developed by Claude Shannon (ell 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? Inputs Output Hot Light Z Z Switch Switch Principles of Computer rchitecture by M. Murdocca and V. Heuring 999 M. Murdocca and V. Heuring

15 Electrically Operated Switch Example: a relay source: UMC, CMSC33, Richard Chang

16 Semiconductors Electrical properties of silicon Doping: adding impurities to silicon Diodes and the P-N junction Field-effect transistors UMC, CMSC33, Richard Chang

17 n Inverter using MOSFET CMOS = complementary metal oxide semiconductor P-type transistor conducts when gate is low N-type transistor conducts when gate is high p-type MOSFET n-type MOSFET UMC, CMSC33, Richard Chang

18 NND GTE

19 NOR GTE

20 CMOS Logic vs ipolar Logic MOSFET transistors are easier to miniaturie CMOS logic has lower current drain CMOS logic is easier to manufacture UMC, CMSC33, Richard Chang

21 CMSC 33, Computer Organiation & ssembly Language Programming, section Fall 23 Homework 3 Due: October 3, 23. (2 points) Draw schematics for the following functions using ND, OR and NOT gates. (Do not simplify the formulas.) (a) X(Y + Z) (b) X + Y Z (c) X(Y + Z) (d) W(X + Y Z) 2. ( points) Question.3, page 493, Murdocca & Heuring 3. ( points) Prove the Consensus Theorem + C + C = + C using the postulates and theorems of oolean algebra (except the Consensus Theorem itself) in Table - (p. 45). Hint: use absorption creatively. 4. (4 points) For each CMOS circuit below, (a) Provide a truth table for the circuit s function. (b) For diagram (a), write down the Sum-of-Products (SOP) oolean formula for the truth table. For diagram (b), wrtie down the Product-of-Sums (POS) oolean formula. (c) Simplify the SOP or POS formula using the postulates and theorems of oolean lgebra (p. 45). Show all work. (d) Draw the logic diagram of the simplified formula using ND, OR, NND, NOR and NOT gates. C D C D (a) (b)

22 Circuits for ddition Next time UMC, CMSC33, Richard Chang

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