Boolean Algebra And Logic Gates Press Enter!!! Chapter 3

Contents:

Contents Introduction Basic Definition Definition of Boolean Algebra Two-valued Boolean Algebra Basic Properties and Theorems of B oolean Algebra Venn Diagram Boolean Function Simplification of Boolean Expressions Canonical and Standard Forms Other Logic Operators Digital Logic Gates Positive and Negative Logic

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Introduction:

Introduction Binary logic deals with variables that have two discrete values 1 – TRUE 0 – FALSE A simple switching circuit containing active elements such as a diode and transistor can demonstrate the binary logic, which can either be ON (switch closed) or OFF (switch open). Electrical signals such as voltage and current exist in the digital system in either one of the two recognized values, except during transition. Switching Algebra or Boolean Algebra Deals with the rules by which logical operations are carried out. Invented by the mathematician George Boole in 19854. Back

Basic Definition:

Basic Definition Boolean Algebra = a set of elements, a set of operators, and a number of assumptions and postulates. A set of elements means any collection of objects having common properties. S denotes a set, X and Y are certain object X ∊ S = X is an object of set S Y ∊ S = Y is not the object of set S A binary operator defined on set S of elements is a rule that assigns to each pair of elements from S a unique elements from S . X*Y = Z – * is a binary operator if all X,Y and Z are of the same set S, else the * can not be binary operator. Back

Definition of Boolean Algebra:

Definition of Boolean Algebra The following Huntington postulates: 1. (a) Closer with respect to the operator (+). (b) Closer with respect to the operator (.) 2. (a) An identity element with respect to + is by 0 A + 0 = 0 + A = A (b) An identity element with respect to . is by 1 A . 1 = 1 . A = A 3. (a) Commutative with respect to (+) A + B = B + A (b) Commutative with respect to (.) A . B = B . A 4. (a) (.) is distributive over (+) A . (B+C) = (A.B) + (A.C) (b) (+) is distributive over (.) A + (B.C) = (A+B) . (A+C) Back

Two-valued Boolean Algebra:

Two-valued Boolean Algebra - is defined on a set of only two elements, S = {0,1}, with rules for two binary operators (+) and (.) and inversion or complement as shown in the following operator tables: Back A B A+B 0 0 0 0 1 1 1 0 1 1 1 1 A B A.B 0 0 0 0 1 0 1 0 0 1 1 1 A A’ 0 1 1 0

Basic Properties and Theorems of Boolean Algebra:

Basic Properties and Theorems of Boolean Algebra 1 . Principle of Duality important property of Boolean algebra means one expression can be obtained from the other in each pair by interchanging every element i.e., every 0 with 1, every 1 with 0, as well as interchanging the operators i.e., every (+) with (.) and every (.) with (+). Back

Venn Diagram:

Venn Diagram is a helpful illustration to visualize the relationship among the variables of a Boolean expression. The diagram consists of rectangle, inside two overlapping circles are drawn, which represent two variables . Back A B AB AB’ A’B

Boolean Function:

Boolean Function is an expression formed with binary variables, the two binary operators AND and OR, one unary operator NOT, parentheses and equal sign Example: F = AB’C F will be 1, A = 1, B = 0, and C = 1 Back A B C F 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1 A F B C F = AB’C

Simplification of Boolean Expressions:

Simplification of Boolean Expressions There are several methods to minimize the Boolean Function. Simplification or minimization of complex algebraic expressions will be shown with the help of postulate and theorems of Boolean algebra. Example 1. Simplify the Boolean function F = AB+BC+B’C Solution: F = AB + BC +B’C =AB + C(B + B’) = AB + C Back

Canonical and Standard Forms:

Canonical and Standard Forms Logical functions – are generally expressed in terms of different combinations of logical variables with their true forms as well as the complement forms. Arbitrary logic function Sum of the Products (SOP) Product of the Sums (POS) Back

Other Logic Operators:

Other Logic Operators Back Boolean Functions Operator Symbol Name Comments F 0 = 0 Null Binary F 1 = A B A . B AND A and B F 2 = A B’ A / B Inhibition A but not B F 3 = A Transfer A F 4 = A’ B B / A Inhibition B but not A F 5 = B Transfer B F 6 = A B’ + A’ B A B Exclusive-OR A or B but not both F 7 = A + B A + B OR A or B F 8 = (A + B)’ A B NOR Not OR F 9 = A B + A’ B’ A B Equivalence * A equals B F 10 = B’ B’ Complement Not B F 11 = A + B’ A B Implication If B then A F 12 = A’ A’ Complement Not A F 13 = A’ + B A B Implication If A then B F 14 = (A B)’ A B NAND Not AND F 15 = 1 Identity Binary constant 1

Digital Logic Gates:

D igital Logic G a tes Tables of Logic gates Extension to Multiple Inputs Universal gates Realization of Logic Functions by NAND Gates Realization of Logic Functions by NOR Gates Two-level Implementation of Logic Networks Multilevel Gating Networks Back

Positive and Negative Logic:

Positive and Negative Logic For a positive logic system, the most positive voltage level represents logic 1 state or HIGH level (H) and the lowest voltage level represents logic 0 state or LOW level (L). For a negative logic system, the most positive voltage level represents logic 0 state and lowest voltage level represents logic 1 state. Back

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Closer Associative Law Commutative Law Identity Element Inverse Distributive Law Various algebraic structures are formulated on the basis of the most common postulates, which are described as follows

Closer:

Closer A set is closed with respect to a binary operator if, for every pair elements of S, the binary operator specifies a rule for obtaining a unique element of S. Back

Associative Law:

A ssociative Law A binary operator * on a set S is said to be associated whenever (A*B)*C = A*(B*C) for all A,B,C ∊ S. Back

Commutative Law:

Commutative Law A binary operator * on a set S is said to be commutative whenever A*B = B*A for all A, B ∊ S. Back

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Identity Element A set S is to have an identity element with respect to a binary operation * on S, if there exists an element E ∊ S with property E*A = A*X = A. Back

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Inverse If a set S has the identity element E with respect to a binary operator * there exists an element B ∊ S, which is called the inverse, for every A ∊ S, such that A*B = E. Back

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Distributive Law If * and (.) are two binary operators on a set S, * is said to be distributive over (.), whenever A*(B.C) = (A*B).(A*C). Back

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The operators and postulates have the following meanings: The binary operator + defines addition. The additive identity is 0. The additive inverse defines subtraction. The binary operator (.) defines multiplication. The multiplication inverse of A is 1/A, defines division. Example: A.1/A = 1 The only distributive law applicable is that of (.) over +. Example: A.(B+C) = (A.B) + (A.C) Back

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5. For every element A,B ∊ S, there exists an element A’ ∊ S (called the complement of A) such that A + A’ = 1 and A . A’ = 0. 6. There exists at least two elements A,B ∊ S, such that A is not equal to B. It is important to realize that in order to have Boolean algebra, the following must be shown: The elements of the set S. The rules of operation for the two binary operators. The set of elements S, together with the two operators satisfies six Hutington postulates. Back

The 5 Postulates:

The 5 Postulates 1. Closure is obviously valid, as form the table it is observed that the result of each operation is either 0 or 1 and 0, 1 ∊ S. 2. From the tables, we can see that ( i ) 0 + 0 = 0 0 + 1 = 1 + 0 = 1 (ii) 1 . 1 = 1 0 . 1 = 1 . 0 = 0 which verifies the two identity elements 0 for (+) and 1 for (.) as defined by postulate 2. 3. The commutative laws are confirmed by the symmetry of binary operator tables. 4. The distributive laws of (.) over (+) can be shown to be applicant with the help of the truth tables considering all the possible values of A, B, and C as under. 5. Postulate 6 also satisfies two-valued Boolean algebra that has distinct elements 0 and 1 where 0 is not equal to 1. Back Click the truth tables to show the table…

2. DeMorgan’s Theorem Two Theorem The complement of a product is equal to the sum of the complements. (A . B)’ = A’ + B’ The complement of a sum is equal to the product of the complements. (A + B)’ = A’ . B’

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3. Other Important Theorems Theorem 1 (a) : A + A = A Theorem 2 (a) : A + 1 = 1 A + A = (A + A) . 1 = (A + A) . (A + A’) = A + A . A’ = A + 0 = A Theorem 1 (b) : A . A = A Theorem 2 (b) : A . 0 = 0 A . A = (A . A) + 0 = (A . A) + (A + A’) = A (A + A’) = A . 1 = A Theorem 3 (a) : A + A . B = A Theorem 3 (b) : A (A + B) = A A + A . B = A . 1 + A . B = A (1 + B) = A . 1 = A

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Complete list of postulates and theorems Postulate 2 (a) A + 0 = A (b) A . 1 = A Postulate 5 (a) A + A’ = A (b) A . A’ = 0 Theorem 1 (a) A + A = A (b) A . A = A Theorem 2 (a) A + 1 = 1 (b) A . 0 = 0 Theorem 3, Involution (A’)’ = A Theorem 3, Commutative (a) A + B = B + A (b) A . B = B . A Theorem 4, Associative (a) A + (B + C)=(A + B) + C (b) A . (B . C) =(A . B) . C Theorem 6, Distributive (a) A (B + C) = A . B + A . C (b) A + B . C=(A + B) . (A + C) Theorem 5, DeMorgan (a) (A + B)’ = A’ . B’ (b) (A . B)’= A’ + B’ Theorem 6, Absorption (a) A + A . B = A (b) A . (A + B) = A Back

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Product Term The logical product of several variables on which a function depends is considered to be a product term. In other words, the AND function is referred to as a product term or standard product. The variables in a product term can be either in true form or in complemented form. ABC’ ---- product term. Sum Term An OR function is referred to as a sum term. The logical sum of several variables on which a function depends is considered to be a sum term. Variables in a sum term can also be either in true form or in complemented form. A+B+C’ ----- sum term Sum of Products (SOP) The logical sum of two or more logical product terms is referred to as a sum of products expression. It is basically an OR operation on AND operated variables. Y=AB+BC+AC ---- sum of products Y=A’B + BC + AC’ ---- sum of products

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Product of Sums (POS) Similarly, the logical product of two or more logical sum terms is called a product of sums expression. It is an AND operation on OR operated variables. Y= (A + B + C) (A + B + C’) ---- product of sums Y= (A’ + B + C) (A’ + B’ + C’) ---- product of sums Standard form The standard form of the Boolean function is when it is expressed in sum of the products or product of the sums fashion. Y= AB +BC +AC ---- standard forms Y= (A + B + C)(A + B’ + C’) ---- standard forms

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Minterm A product term containing all n variables of the function in either true or complement form. Each minterm is obtained by an AND operation of the variables in their true form or complemented form. A B C Minterm 0 0 0 A’B’C’ 0 0 1 A’B’C 0 1 0 A’BC’ 0 1 1 A’BC 1 0 0 AB’C’ 1 0 1 AB’C 1 1 0 ABC’ 1 1 1 ABC

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Canonical Sum of Product Expression When a Boolean function is expressed as the logical sum of all the minterms from the rows of a truth table, for which the value of the function is 1. Example: F (A, B) = A + B = A.1 + B.1 = A (B + B’) + B (A + A’) = AB + AB’ + AB + A’B F (A, B) = AB + AB’ + A’B (as AB + AB = AB)

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Maxterm A sum term containing all n variables of the function in either true or complemented form. A B C Maxterm 0 0 0 A + B + C 0 0 1 A + B + C’ 0 1 0 A + B’ + C 0 1 1 A + B’ + C’ 1 0 0 A’ + B + C 1 0 1 A’ + B + C’ 1 1 0 A’ + B’ + C 1 1 1 A’ + B’ + C’

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Canonical Product of Product Expression When a Boolean function is expressed as the logical product of all the maxterms from the rows of a truth table, for which the value of the function is 0. Example: F (A, B, C) = (A + B’) (B + C) (A + C’) = (A + B’ + 0) (B + C + 0) (A + C’ + 0) = (A + B’ + CC’) (B + C + AA’) (A + C’ + BB’) = (A + B’ + C) (A + B’ + C’) (A + B + C) (A’ + B + C) (A + B + C’) (A + B’ + C’) F (A, B, C) = (A + B’ + C) (A + B’ + C’) (A + B + C) (A’ + B + C) (A + B + C’)

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Sum of Products (SOP) The sum of products (SOP) expression of a Boolean function can be obtained from its truth table, performing OR operation of he product terms corresponding to the combinations containing a function value of 1. Y = A’BC’ + AB’C’ + AB’C + ABC’ Inputs Output Y Product terms Sum terms A B C 0 0 0 0 A + B + C 0 0 1 0 A + B + C’ 0 1 0 1 A’BC’ 0 1 1 0 A + B’ + C’ 1 0 0 1 AB’C’ 1 0 1 1 AB’C 1 1 0 1 ABC’ 1 1 1 0 A’ + B’ + C’

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Product of Sums (POS) The product of sums (POS) expression of a Boolean function can be obtained from its truth table, performing AND operation of the sum terms corresponding to the combinations containing a function value of 0. Y = (A+B+C) ( A+B+C’) ( A+B’+C’) (A’+B’+C’) Inputs Output Y Product terms Sum terms A B C 0 0 0 0 A + B + C 0 0 1 0 A + B + C’ 0 1 0 1 A’BC’ 0 1 1 0 A + B’ + C’ 1 0 0 1 AB’C’ 1 0 1 1 AB’C 1 1 0 1 ABC’ 1 1 1 0 A’ + B’ + C’ Back

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Logic Value Signal Value Logic Value Signal Value 1 H 0 H 0 L 1 L positive logic system negative logic system IC family types Supply Voltage (V) High-level voltage (V) Low-level voltage (V) Range Typical Range Typical TTL V cc = 5 2.4 to 5 3.5 0 to 0.4 0.2 ECL V EE = -5.2 -0.95 to – 0.7 -0.8 -1.9 to -1.6 -1.8 CMOS V DD = 3 to 10 V DD V DD 0 to 0.5 0 Positive logic Logic 1 Logic 0 Negative logic Logic 0 Logic 1 Back

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Basic Logic Gates Name Graphic Symbol Algebraic Function Truth Table A B F AND F = AB 0 0 0 1 1 0 1 0 0 0 1 OR F = A+B 0 0 0 1 1 0 1 0 1 1 1 Inverter or NOT F = A’ 0 1 1 0

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Other Logic Gates Name Graphic Symbol Algebraic Function Truth Table A B F Buffer F = A 0 1 0 1 NAND F = (AB)’ 0 0 0 1 1 0 1 1 1 1 0 NOR F = (A + B)’ 0 0 0 1 1 0 1 1 0 0 0 Exclusive-OR (XOR) F = AB’ + A’B = A B 0 0 0 1 1 0 1 0 1 1 0 Equivalence or Exclusive-NOR (XNOR) F = AB + A’B = A B 0 0 0 1 1 0 1 1 0 0 1 Back

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Extension to Multiple Inputs AND and OR gates are both commutative and associative. A F = ABC B C = A F = ABC B C

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The NAND and NOR functions are the complement of AND and OR functions respectively. They are commutative, but not associative. = A F = (A+B+C)’ = A’B’C’ B C A F = (A+B)C’ B C

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The Exclusive-Or (XOR) gates and equivalence gates both possess commutative and associative properties, and they can extended to multiple input variables. Equivalence gate or XNOR gate is equivalent to XOR gate followed by NOT gate and hence its logic behavior is opposite to the XOR gate. A F = (A+B)C’ B C = A F = A B C B C Back

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Universal Gates NAND gates and NOR gates are called universal gates or universal building blocks, as any type of gates or logic functions can be implemented by these gates. NOT function: F = A’ AND function: F = AB A F = A’ A F = AB B (AB)’ Back

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Realization of Logic Functions by NAND Gates To achieve the realization of logic functions by NAND gates, first step is to express the function in SOP form and simply replace the gates with NAND gates. F = AB + CD A F B C D F A B C D Back

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Realization of Logic Functions by NOR Gates To achieve the realization of logic functions by NOR gates only, the first step is to express the function at POS form and replace the AND gates and OR gates with NOR gates. F = (A+B) (C+D) F A B C D F A B C D Back

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Two-level Implementation of Logic Networks The maximum number of gates cascaded in series between an input and output is called the level of gates. The SOP expression can be implemented using a two-level gate network, AND gates at the first-level and OR gates at the second-level. The POS expression can be implemented using a two-level gate network, OR gates at the first-level and AND gates at the second-level. F F A B C D A B C D Back

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Multilevel Gating Networks If a switching network is implemented using gates in more than two levels. F = BC’ + A’B + D F B C’ B A’ F A’ C’ B D D Consists of two 2-input AND gates, a 3-input OR gate, and five literals or inputs. A three-level representation of the same function containing two 2-input AND gate and four literals. Back

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1. A binary operator * on a set S is said to be commutative (A*B=B*A). a. Associative Law b. Inverse c. Commutative Law d. Element Click your correct answer

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2. A set of S to have an element with respect to a binary operator* on S, if there exists an element E ∊ S with property. a. A*B=B*A b. A*(B.C) c. Identity Element d. Distribution Law Click your correct answer

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3. If * and (.) are the two binary operators on a set S. a. Commutative Law b. Morgan Law c. Distribution Law d. Associative Law Click your correct answer

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4. Who introduced the systematic approach of logic and developed and algebraic system? a. DeMorgan b. Huntington c . George Boole d. Boolean Click your correct answer

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5. The Tabular form of the values of a Boolean function according to the all possible values of it’s variables. a. Venn diagram b. Product term c . Truth table d. Boolean function Click your correct answer

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6. The expression of a Boolean function that obtained from it’s truth table summing or performing OR operation of the product terms corresponding to the combination containing a function value a. Sum term b . Sum of products c. Product of sums d. Product term Click your correct answer

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7. It expressed the terms of AND,OR, and NOT operations. a . Boolean function b. Boolean expression c. DeMorgan’s theorem d. Sum of product expression Click your correct answer

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8. Boolean function is expressed as the logical product of all the maxterms from the rows of a truth table, and they expressed. a. Canonical sum of product expression b. Standard form c. Logic function d. Canonical product of sum expression Click your correct answer

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9. A product term containing all n variables of the function in either true or complemented form. a. Minterm b. Maxterm c. Sum term d. Standard form Click your correct answer

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10. A sum term containing all n variables of the function in either true or complemented form. a. Minterm b . Maxterm c. Sum term d. Product term Click your correct answer

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11. It is very much similar to algebra of some respects. a. Boolean function b. Boolean expression c. Ordinary algebra d . Boolean algebra Click your correct answer

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12. Helpful illustration to visualize the relationship among the variables of a Boolean implemented a. DeMorgan’s theorem b. Venn diagram c. Boolean expression implemented d. Deriving a product of sums expression Click your correct answer

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13. The maximum number of gates cascaded in series between an input and output a. Generate form b. Level of gates c. Gates d. Level of form Click your correct answer

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14. It is generally expressed in terms of different combinations of logical variables with their true forms. a. Boolean function b. Logical function c. Logic function d. Distributive law Click your correct answer

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15. The most positive voltage level represents in logic. a. Negative logic b . Positive logic c. Logic function d. Logical function Click your correct answer

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16. The most positive voltage level represents in logic or states and lowest voltage level represents in logic. a . Negative logic b. Positive logic c. Logical function d. Logic function Click your correct answer

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17. What the name of this graphic symbol? a. NAND b. AND c. Buffer d. OR Click your correct answer

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18. What the name of this graphic symbol? a. NOT b. NOR c. XOR d. XNOR Click your correct answer

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19. Give the correct equation of this. a. F= A’ + B’ + C’ b. F = ABC c. F = A’B’C’ d. F = A + B + C A B C F Click your correct answer

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20. Give the correct equation of this. a. F= A’ + B’ + C’ b. F = ABC c. F = A’B’C’ d. F = A + B + C A’ B’ C’ F Click your correct answer

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THANK YOU VERY MUCH BY: OLIVER A. VILLASIS

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