# Chapter 2: Boolean Algebra and Logic Gates

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1 Chapter 2: Boolean Algebra and Logic Gates Mathematical methods that simplify binary logics or circuits rely primarily on Boolean algebra. Boolean algebra: a set of elements, a set of operators, and a number of unproved axioms or postulates. A set of elements is any collection of objects, usually having a common property. A = {1, 2, 3, 4} indicates that set A has the elements of 1, 2, 3, and 4. A binary operator defined on a set S of elements is a rule that assigns, to each pair of elements from S, a unique element from S. The most common postulates used to formulate various algebraic structures are as follows: 1.Closure. A set S is closed with respect to a binary operator if, for every pair of elements of S, the binary operator specifies a rule for obtaining a unique element of S. 2.Associative law. A binary operator * on a set S is said to be associative whenever (x * y) * z = x * (y * z) for all x, y, z, S 3.Commutative law. A binary operator * on a set S is said to be commutative whenever x * y = y * x for all x, y S NCNU_2016_DD_2_1

2 4.Identity element. A set S is said to have an identity element with respect to a binary operation * on S if there exists an element e S with the property that e * x = x * e = x for every x S Example: The element 0 is an identity element with respect to the binary operator + on the set of integers I = {, -3, -2, -1, 0, 1, 2, 3, }, since x + 0 = 0 + x = x for any x I The set of natural numbers, N, has no identity element, since 0 is excluded from the set. 5.Inverse. A set S having the identity element e with respect to a binary operator * is said to have an inverse whenever, for every x S, there exists an element y S such that x * y = e Example: In the set of integers, I, and the operator +, with e = 0, the inverse of an element a is (-a), since a + (-a) = 0. 6.Distributive law. If * and are two binary operators on a set S, * is said to be distributive over whenever x * (y z) = (x * y) (x * z) NCNU_2016_DD_2_2

3 Field A field is an example of an algebraic structure. The field of real numbers is the basis for arithmetic and ordinary algebra. The binary operator + defines addition. The additive identity is 0. The additive inverse defines subtraction. The binary operator defines multiplication. The multiplicative identity is 1. For a 0, the multiplicative inverse of a = 1/a defines division (i.e., a 1/a = 1). The only distributive law applicable is that of over +: a (b + c) = (a b) + (a c) NCNU_2016_DD_2_3

4 Axiomatic Definition of Boolean Algebra 1854: George Boole developed an algebraic system now called Boolean algebra. 1904: E. V. Huntington formulated a set of postulates that formally define the Boolean algebra 1938: C. E. Shannon introduced a two-valued Boolean algebra called switching algebra that represented the properties of bistable electrical switching circuits Two binary operators, + and, (Huntington) postulates: 1. (a) The structure is closed with respect to the operator +. (b) The structure is closed with respect to the operator. 2. (a) The element 0 is an identity element with respect to +; that is, x + 0 = 0 + x = x. (b) The element 1 is an identity element with respect to ; that is, x 1 = 1 x = x. 3. (a) The structure is commutative with respect to +; that is, x + y = y + x. (b) The structure is commutative with respect to ; that is, x y = y x. 4. (a) The operator is distributive over +; that is, x (y + z) = (x y) + (x z). (b) The operator + is distributive over ; that is, x + (y z) = (x + y) (x + z). NCNU_2016_DD_2_4

5 5. For every element x B, there exists an element x B (called the complement of x) such that (a) x + x = 1 and (b) x x = There exist at least two elements x, y B such that x y. Comparing Boolean algebra with arithmetic and ordinary algebra 1. Huntington postulates do not include the associative law. However, this law holds for Boolean algebra and can be derived (for both operators) from the other postulates. 2. The distributive law of + over (i.e., x + (y z) = (x + y) (x + z) ) is valid for Boolean algebra, but not for ordinary algebra. 3. Boolean algebra does not have additive or multiplicative inverses; therefore, there are no subtraction or division operations. 4. Postulate 5 defines an operator called the complement that is not available in ordinary algebra. 5. Ordinary algebra deals with the real numbers, which constitute an infinite set of elements. Boolean algebra deals with the as yet undefined set of elements, B, but in the two valued Boolean algebra defined next (and of interest in our subsequent use of that algebra), B is defined as a set with only two elements, 0 and 1. NCNU_2016_DD_2_5

6 Two-valued Boolean Algebra NOT also called INV B = {0,1} The rules of operations AND OR NOT x y x y x y x+y x x OR gate performs Boolean addition AND gate performs Boolean multiplication Closure: the result of each operation is either 1 or 0 and 1, 0 B. Identity elements: 0 for + and 1 for The commutative laws are obvious from the symmetry of the binary operator tables. NCNU_2016_DD_2_6

7 Distributive laws: x (y + z) = (x y) + (x z) x+ (y z) = (x+y) (x+z) y z x+(y z) x+y x+z (x+y) (x+z) x y z x y z f 1 x y x z x f 2 y x f 2 z f1 NCNU_2016_DD_2_7

8 Complement x+x'=1: 0+0'=0+1=1; 1+1'=1+0=1 x x'=0: 0 0'=0 1=0; 1 1'=1 0=0 Has two distinct elements 1 and 0, with 0 1 We have just established a two valued Boolean algebra: a set of two elements + : OR operation; : AND operation a complement operator: NOT operation x or x Binary logic is a two-valued Boolean algebra also called switching algebra by engineers NCNU_2016_DD_2_8

9 Basic Theorems and Properties of Boolean Algebra Duality the binary operators are interchanged; AND OR the identity elements are interchanged; 1 0 NCNU_2016_DD_2_9

10 Theorem 1(a): x + x = x x + x = (x + x) 1 by postulate: 2(b) = (x + x) (x + x') 5(a) = x + xx' 4(b) = x + 0 5(b) = x 2(a) Theorem 1(b): x x = x x x = x x + 0 by postulate: 2(a) = xx + xx' 5(b) = x (x + x') 4(a) = x 1 5(a) = x 2(b) Theorem 1(b) is the dual of theorem 1(a) NCNU_2016_DD_2_10

11 Theorem 2(a): x + 1 = 1 x + 1 = 1 (x + 1) by postulate: 2(b) = (x + x')(x + 1) 5(a) = x + x' 1 4(b) = x + x' 2(b) = 1 5(a) Theorem 2(b): x 0 = 0 by duality Theorem 3: (x')' = x Postulate 5 defines the complement of x, x + x' = 1 and x x' = 0 The complement of x' is x is also (x')' NCNU_2016_DD_2_11

12 Theorem 6(a): x + xy = x x + xy = x 1 + xy by postulate: 2(b) = x (1 +y) 4(a) = x 1 2(a) = x 2(b) Theorem 6(b): x (x + y) = x by duality By means of truth table x y xy x + xy x y x y x x NCNU_2016_DD_2_12

13 DeMorgan's Theorems (x+y)' = x' y' x y x+y (x+y) x y x y (x y)' = x' + y' x y xy (xy) x y x +y NCNU_2016_DD_2_13

14 Some Simple Applications of Logic seat belt alarm level indicator intrusion detection landing gear status NCNU_2016_DD_2_14

15 Operator Precedence The operator precedence for evaluating Boolean expressions is 1. parentheses 2. NOT 3. AND 4. OR Examples f 1 = xy'+ z (x y + z)' x y z f 1 x y z f 2 NCNU_2016_DD_2_15

16 Boolean Functions A Boolean function is an algebraic expression consists of binary variables binary operators OR and AND unary operator NOT parentheses A Boolean function expresses the logical relationship between binary variables and is evaluated by determining the binary value of the expression for all possible values of the variables. Examples F 1 = x + y z F 1 = 1 if x = 1 or if y = 1 and z = 0, others F 1 = 0. F 2 = x' y' z + x' y z + x y F 2 = 1 if (x = 0, y = 0, z = 1) or (x = 0, y = 1, z = 1) or (x = 1, y = 0), others F 2 = 0. What are the others? NCNU_2016_DD_2_16

17 Truth Table Boolean function can be represented in a truth table. Truth table has 2 n rows where n is the number of variables in the function. The binary combinations for the truth table are obtained from the binary numbers by counting from 0 through 2 n -1. symbol of logic gates AND OR NOT Implementation of F 1 with logic gates F 1 = x + y z NCNU_2016_DD_2_17

18 Equivalent Logics Boolean function can be represented in truth table only in one way. In algebraic form, it can be expressed in a variety of ways, all of which have equivalent logic. Using Boolean algebra, it is possible to obtain a simpler expression for the same function with less number of gates and inputs to the gate. Designers work on reducing the complexity and number of gates to significantly reduce the circuit cost. F 2 = x y z + x yz + xy = x z(y + y) + xy = x z + xy NCNU_2016_DD_2_18

19 Algebraic Manipulation To minimize Boolean expressions literal: a complemented or un-complemented variable (an input to a gate) term: an implementation with a gate The minimization of the number of literals and the number of terms => a circuit with less logic gates F 2 = x y z + x yz + xy 3 terms, 8 literals = x z(y + y) + xy = x z + xy 2 terms, 4 literals Functions of up to five variables can be simplified by the map method described in the next chapter. For complex Boolean functions and many different outputs, designers of digital circuits use computer minimization programs that are capable of producing optimal circuits with millions of logic gates. NCNU_2016_DD_2_19

20 Minimization of Boolean Function NCNU_2016_DD_2_20

21 Complement of a Function F is obtained from an interchange of 0's for 1's and 1's for 0's in the value of F The complement of a function may be derived using DeMorgan's theorem. Three-variable DeMorgan's theorem: (A + B + C) = (A + X) let B + C = X = A X by DeMorgan's = A (B + C) X = B + C = A (B C ) by DeMorgan's = A B C associative Generalized form (A + B + C F) = A B C... F (ABC... F) = A + B + C F this is very useful NCNU_2016_DD_2_21

22 EXAMPLE 2.2 EXAMPLE 2.3 NCNU_2016_DD_2_22

23 Minterms and Maxterms A minterm (standard product): an AND term consists of all literals in their normal form or in their complement form For example, two binary variables x and y, has 4 minterms xy, xy', x'y, x'y' n variables can be combined to form 2 n minterms (m j, j = 0 ~ 2 n -1) A maxterm (standard sum): an OR term; 2 n maxterms (M j, j = 0 ~ 2 n -1) Each maxterm is the complement of its corresponding minterm, and vice versa. NCNU_2016_DD_2_23

24 Canonical Form: Sum of Minterms An Boolean function can be expressed by a truth table x y z m 1 sum of minterms f = Σ m j product of maxterms f = Π M j f 1 = x'y'z + xy'z' + xyz = m 1 + m 4 +m 7 f 2 = x'yz + xy'z + xyz' + xyz = m 3 + m 5 +m 6 + m 7 x y z x y z m 4 m 7 f 1 m 7 m 1 m 3 m 4 m 7 m 5 m 6 m 7 NCNU_2016_DD_2_24

25 Canonical Form: Product of Maxterms The complement of a Boolean function the minterms that produce a 0 f 1 ' = m 0 + m 2 +m 3 + m 5 + m 6 = x'y'z' + x'yz' + x'yz + xy'z + xyz' f 1 = (f 1 ')' = (x + y + z)(x + y' + z)(x + y' + z')(x' + y + z')(x' + y' + z) = M 0 M 2 M 3 M 5 M 6 f 2 = (x + y + z)(x + y + z')(x + y' + z)(x' + y + z) = M 0 M 1 M 2 M 4 Canonical form: any Boolean function expressed as a sum of minterms or a product of maxterms M 0 M 2 M 3 M 5 M 6 M 0 M 1 M 2 M 4 NCNU_2016_DD_2_25

26 Minterm Expansion EXAMPLE 2.4: Express the Boolean function F = A + B'C as a sum of minterms. F = A + B'C = A (B + B') + B'C = AB + AB' + B'C = AB(C + C') + AB'(C + C') + (A + A')B'C = ABC + ABC' + AB'C + AB'C' + A'B'C = A'B'C + AB'C' + AB'C + ABC' + ABC = m 1 + m 4 + m 5 + m 6 + m 7 F(A,B,C) = Σ (1, 4, 5, 6, 7) or, built the truth table first m 1 (A B C) m 4 (A B C ) m 5 (A B C) m 6 (A B C ) m 7 (A B C) NCNU_2016_DD_2_26

27 Maxterm Expansion EXAMPLE 2.5: Express the Boolean function F = xy + x'z as a product of maxterms. F = xy + x'z= (xy + x') (xy + z) = (x + x')(y + x')(x + z)(y + z) = (x' + y)(x + z)(y + z) x' + y = x' + y + zz' = (x' + y + z)(x' + y + z') x + z = x + z + yy' = (x + y + z)(x + y' +z) y + z = y + z + xx' = (x + y + z)(x' + y +z) F = (x + y + z)(x + y' + z)(x' + y + z)(x' + y + z') = M 0 M 2 M 4 M 5 F(x,y,z) = Π (0,2,4,5) check this result with truth table NCNU_2016_DD_2_27

28 Canonical Form Conversion Conversion between Canonical Forms F(A,B,C) = (1,4,5,6,7) F (A,B,C) = (0,2,3) = m 0 + m 2 + m 3 By DeMorgan's theorem F = (F ) = (m 0 + m 2 + m 3 ) = m 0 m 2 m 3 = M 0 M 2 M 3 = Π(0, 2, 3) m j ' = M j sum of minterms product of maxterms interchange the symbols and and list those numbers missing from the original form of 1's of 0's NCNU_2016_DD_2_28

29 Conversion Example F = xy + x z F(x, y, z) = (1, 3, 6, 7) F(x, y, z) = (0, 2, 4, 5) NCNU_2016_DD_2_29

30 Standard Forms Canonical forms are baseline expression and seldom used, they are not minimum Two standard forms are used usually sum of products (SOP) F 1 = y' + xy + x'yz' product of sums (POS) F 2 = x(y' + z)(x' + y + z') This circuit configuration is referred to as a two level implementation. In general, a two level implementation is preferred because it produces the least amount of delay through the gates when the signal propagates from the inputs to the output. However, the number of inputs to a given gate might not be practical. NCNU_2016_DD_2_30

31 Nonstandard Forms F 3 = AB + C(D + E) = AB + C(D + E) = AB + CD + CE Which kind of gate will have the least delay (high switching speed)? The delay through a gate is largely dependent on the circuit design and technology, as well as manufacturing process used. (will be taught in VLSI design) NCNU_2016_DD_2_31

32 Other Logic Operations 2 n rows in the truth table of n binary variables 2 2n functions for n binary variables (each row may either be 0 or 1) 16 (2 22 )functions of two binary variables NCNU_2016_DD_2_32

33 commonly used logic gates NCNU_2016_DD_2_33

34 Digital Logic Gates of Two Inputs NCNU_2016_DD_2_34

35 Digital Logic Gates of Two Inputs NCNU_2016_DD_2_35

36 Extension to Multiple Inputs A gate can be extended to multiple inputs if its binary operation is commutative and associative AND and OR operations are commutative and associative commutative: x + y = y + x, xy = yx associative: (x + y) + z = x + (y + z) = x + y + z, (x y)z = x(y z) = x y z x y z F x y z F x y z F x y z F NCNU_2016_DD_2_36

37 Multiple-input NOR/NAND NAND and NOR are commutative but not associative => they are not extendable (x y) z = [(x + y) + z] = (x + y) z = xz + yz x (y z) = [x + (y + z) ] = x (y + z) = x y + x z NCNU_2016_DD_2_37

38 Multiple-input NOR/NAND Multiple-input NOR = a complement of OR gate (x y z) = (x + y + z) Multiple-input NAND = a complement of AND (x y z) = (xyz) The cascaded NAND operations = sum of products The cascaded NOR operations = product of sums DeMorgan s theorems are useful here. NCNU_2016_DD_2_38

39 Multiple-input XOR/XNOR The XOR and XNOR gates are commutative and associative Multiple-input XOR gates are uncommon (nowadays this is not true anymore!) XOR(XNOR) is an odd(even) function: it is equal to 1 if the inputs variables have an odd(even) number of 1's NCNU_2016_DD_2_39

40 Positive and Negative Logic* Two signal values (High/Low) <=> two logic values (1/0) positive logic: H = 1; L = 0 negative logic: H = 0; L = 1 Positive logic is commonly used. NCNU_2016_DD_2_40

41 Digital Logic Families Digital circuit technology (off the shelf): TTL: transistor-transistor logic (fading out?) ECL: emitter-coupled logic (high speed, high power consumption) MOS: metal-oxide semiconductor (NMOS, high density) CMOS: complementary MOS (low power) CMOS technology now dominates the main stream of IC design, it will be taught in Introduction to VLSI Design course. TTL 7400 Quad 2-input NAND CMOS 4001 Quad 2-input NOR NCNU_2016_DD_2_41

42 NCNU_2016_DD_2_42

43 Some Important Parameters of Logic Families* Fan out specifies the number of standard loads that the output of a typical gate can drive without impairing its normal operation. A standard load is usually defined as the amount of current needed by an input of another similar gate in the same family. Fan in is the number of inputs available in a gate. Power dissipation is the power consumed by the gate that must be available from the power supply. Propagation delay is the average transition delay time for a signal to propagate from input to output. For example, if the input of an inverter switches from 0 to 1, the output will switch from 1 to 0, but after a time determined by the propagation delay of the device. The operating speed is inversely proportional to the propagation delay. Noise margin is the maximum external noise voltage added to an input signal that does not cause an undesirable change in the circuit output. NCNU_2016_DD_2_43

44 Homework #2 from Mano. Due: one week. 2.2 Simplify the following Boolean expressions to a minimum number of literals: (a) x y + x y (b) (x + y) (x + y ) (e) (a + b + c )(a b + c) (f) a b c + ab c + abc + a bc 2.4 Reduce the following Boolean expressions to the indicated number of literals: (a) A C + ABC + AC to three literals (c) A B(D + C D) + B(A + A CD) to one literal (d) (A + C) (A + C ) (A + B + C D) to four literals 2.9 Find the complement of the following expressions: (a) x y + xy (c) z + z (v w + xy) 2.11 List the truth table of the function: (a) F = y z + x y + xz + x z (b) F = ac + bc 2.14 Implement the Boolean function: F = x y + x z + xy (a) With AND, OR, and inverter gates (b) With OR and inverter gates (d) With NAND and inverter gates NCNU_2016_DD_2_44

45 2.22 Convert each of the following expressions into sum of products and product of sums: (a) (u + x w)(x + u v) (b) x + z(x + y )(y + z') 2.30 Write the following Boolean expressions in sum of products form: (b + d)(a + b + c)(a + c) 2.31 Write the following Boolean expression in product of sums form: a b + a c + bc NCNU_2016_DD_2_45

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