1 / 31

R. Johnsonbaugh Discrete Mathematics 5 th edition, 2001

R. Johnsonbaugh Discrete Mathematics 5 th edition, 2001. Chapter 9 Boolean Algebras and Combinatorial Circuits. A bit is a 0 or a 1. Input x 1 , x 2 can be 0 or 1 ------------------------------------------- Conjunction: AND x 1  x 2 = 1 if x 1 = x 2 = 1

owenolsen
Download Presentation

R. Johnsonbaugh Discrete Mathematics 5 th edition, 2001

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. R. JohnsonbaughDiscrete Mathematics5th edition, 2001 Chapter 9 Boolean Algebras and Combinatorial Circuits

  2. A bit is a 0 or a 1. Input x1, x2 can be 0 or 1 ------------------------------------------- Conjunction: AND x1 x2 = 1 if x1 = x2 = 1 0 otherwise Logical gates 9.1 Combinatorial circuits

  3. Inclusive disjunction OR x1 v x2 = 1 if x1 = 1 or x2 = 1 0 otherwise

  4. NOT ~ x = 1 if x = 0 0 if x = 1 Negation

  5. A logic table of a combinatorial circuit is a table with all possible inputs and all its resulting outputs Logic Tables

  6. A Boolean expression such as y = ~((x1^x2)vx3) is composed of literals like x1, x2, x3 and symbols like ^, v, ~, ) and ( We write y = y(x1, x2, x3) The value of a Boolean expression is the bit obtained when the literals are replaced by bits, e.g. y(0,1,0) = 1. Boolean expressions

  7. Combinatorial circuits

  8. The logic table corresponding to the Boolean expression y = ~((x1 ^ x2) v x3). Logic table

  9. 4. Identity laws av0 = a a^1 = a 5. Complement laws av(~a) = 1 a^(~a) = 0 For a, b, c  {0, 1} 1. Associative laws av(bvc) = (avb)vc a^(b^c) = (a^b)^c 2. Commutative laws avb = bva a^b = b^a 3. Distributive laws a^(bvc) = (a^b) v (a^c) av(b^c) = (avb) ^ (avc) 9.2 Properties of Combinatorial Circuits

  10. Boolean expressions Two Boolean expressions X = X(x1, x2,…,xn) and Y = Y(x1, x2,…, xn) are equal (write X = Y) if and only if X(a1,a2,…, an) = Y(a1, a2,…, an) for all ai{0, 1}, 1 < i < n. • Example: X(x1,x2, x3) = ~x1^(x2vx3) and Y(x1,x2, x3) = (~x1^x2) v (~x1^x3) are equal.

  11. Equivalent combinatorial circuits • Two combinatorial circuits, each having inputs x1,…, xn and a single output for each set of inputs, are equivalent if, given the same inputs, the two circuits produce the same output. • Theorem 9.2.7: Let C and D be two combinatorial circuits with X and Y the Boolean expressions that represent C and D, respectively. C and D are equivalent if and only if X = Y,

  12. Let A be a nonempty set containing distinct elements 0 and 1, endowed with two binary operations + and • and a unary operation ~ (complement) that satisfy properties 1 to 5. Then A is called a Boolean algebra. 1. Associative laws a + (b + c) = (a + b) + c a • (b • c) = (a • b) • c 2. Commutative laws a + b = b + a a • b = b • a 3. Distributive laws a • (b + c) = (a • b) + (a • c) a + (b • c) = (a + b) • (a + c) 4. Identity laws a + 0 = a a • 1 = a 5. Complement laws a + (~a) = 1 a • (~a) = 0 9.3 Boolean algebras

  13. Example 1 of a Boolean algebra Example 1: Z2 = {0,1}, with the operations  and  defined by the tables below, is a Boolean algebra.

  14. Let U be a universal set and S = P(U) = power set of U. For any elements X, Y  S define X + Y = X  Y X·Y = X  Y ~X = U - X Let 0 =  and 1 = U. Then properties 1-5 hold and S is a Boolean algebra. Example 2: Power set

  15. Complement in a Boolean algebra • In a Boolean algebra A, given x  A, an element x' such that x' + x = 1 and x' • x = 0 is called the complement of x. Theorem 9.3.4: The complement x' of each x in a Boolean algebra is unique. Specifically, if x + y = 1 and x • y = 0 then y = x'.

  16. Equations in Boolean algebras come in pairs. For example, the identity laws x+0 = x and x•1 = x. These are called dual statements. The dual of a statement in Boolean algebras is obtained by making the following replacements: 0 replaced by 1, 1 replaced by 0, + replaced by •, • replaced by + Dual statements

  17. A duality theorem The following is an important theorem about duality statements in a Boolean algebra: • Theorem 9.3.10: The dual of a theorem about Boolean algebras is also a theorem.

  18. 9.4 Boolean functions and synthesis of circuits Let X(x1,…, xn) be a Boolean expression. • A Boolean function is a function of the form f(x1,…, xn) = X(x1,…, xn) • Its domain is (Z2)n = {(x1, x2,…, xn) |xi {0, 1}, 1 < i < n} and its range is Z2 = {0, 1} X : (Z2)n Z2

  19. Exclusive OR, in symbol  The binary operation  can be expressed as a Boolean function x1 x2 = X(x1 ,x2), with domain = {(1,1),(1,0),(0,1),(0,0)} = (Z2)2; and range = Z2 = {0, 1}. Example of a Boolean function

  20. Minterm A minterm in the symbols x1,…, xn is a Boolean expression of the form m = y1^y2^…^yn where each yk is either xk or its complement xk', 1 < k < n.

  21. Disjunctive normal form Theorem 9.4.6: Let f: (Z2)n Z2 be a Boolean function not identically zero. • Let A1, A2,…, Ak denote the elements of (Z2)n such that f(Ai) = 1. • For each Ai = (a1,…, an) set mi = y1^…^ yn • where yj = xj (if aj = 1) OR ~xj (if aj = 0). • Then f(x1,…, xn) = m1v…v mk. • The disjunctive normal form of a Boolean function is a representation of f as a disjunction of mk's as in the theorem.

  22. Let f(x1, x2) = (~x1)vx2. Its truth table is A1 = (1,1) A2 = (0,1) A3 = (0,0) Then m1 = x1^ x2 m2 = (~x1)^x2 m3 = (~x1)^(~x2) It can be verified that f(x1, x2) = m1 v m2 v m3 Use of disjunctive normal form

  23. Definition of functionally complete set of gates: A gate is a function f : (Z2)n  Z2 , for n > 1 A set of gates S = {g1, g2,…, gk} is said to be functionally complete if given any positive integer n and a function f : (Z2)n  Z2 , it is possible to construct a combinatorial circuit that computes f using only the set of gates S. 9.5 Applications

  24. NAND • The NAND gate x1x2 is equivalent to ~(x1^x2), and is defined by the table and the diagram

  25. NAND circuits • Using NAND we can write • x' = xx • x v y = (xx)(yy)

  26. Functionally complete sets of gates Theorem 9.4.6: The set of gates {AND, OR, NOT} is functionally complete. Theorem 9.5.5: The sets of gates S1 = {AND, NOT} and S2 = {OR, NOT} are functionally complete. Theorem 9.5.7: The set {NAND} is functionally complete.

  27. Minimization problem The minimization problem consists of finding the circuit that represents a Boolean function with the minimal number of gates. • Example: The following circuit represents the Boolean function f(x,y,z) = xyz v xyz' v xy'z' = x(y v z')

  28. Input: two bits x and y Two outputs: binary sum cs, where c is the carry bit and s is the sum bit. Half adder

  29. Half adder circuit Two outputs: • s = (x v y)^[~(x^y)] = x  y • c = x^y

  30. Full adder Full adder table

  31. A circuit to add two 3-bit numbers

More Related