Logic Circuits and Boolean Algebra

Introduction1854: Logical algebra was published by George Boole known today as Boolean AlgebraIts a convenient way and systematic way of expressing and analyzing the operation of logic circuits.1938: Claude Shannon was the first to apply Booles work to the analysis and design of logic circuits.

Boolean Operations & ExpressionsVariable a symbol used to represent a logical quantity.Complement the inverse of a variable and is indicated by a bar over the variable.Literal a variable or the complement of a variable.

LOGIC GATESIn formal logic, a statement (proposition) is a declarative sentence that is either true(1) or false (0). It is easier to communicate with computers using formal logic.

Boolean variable: Any literal symbol that takes only two values either true (1) or false (0) such as x,y,z used to represent an element of B={0,1}. They are used as basic units of formal logic.

Boolean ExpressionBoolean expression: Let x1, x2,,xn be Boolean variables. A Boolean expression over B is defined recursively as follows:0 and 1 are Boolean expressionsx1, x2,,xn are Boolean expressionsIf is a Boolean expression, then is a Boolean expressionIf 1 and 2 are Boolean expressions, then (1 2) and (1 + 2) are Boolean expressions.

Boolean function and logic diagram Boolean function: Mapping from Boolean variables to a Boolean value.

Truth table: Represents relationship between a Boolean function and its binary variables. It enumerates all possible combinations of arguments and the corresponding function values.

Boolean function and logic diagram Boolean algebra: Deals with binary variables and logic operations operating on those variables.

Logic diagram: Composed of graphic symbols for logic gates. A simple circuit sketch that represents inputs and outputs of Boolean functions.

GatesIt refers to the hardware to implement Boolean operators.A gate is a logic circuit with one or more input signals but only one output signal.The most basic gates are

Boolean function and truth table

Boolean AdditionBoolean addition is equivalent to the OR operation

A sum term is produced by an OR operation with no AND ops involved.i.e.A sum term is equal to 1 when one or more of the literals in the term are 1.A sum term is equal to 0 only if each of the literals is 0. 0+0 = 00+1 = 11+0 = 11+1 = 1

Boolean MultiplicationBoolean multiplication is equivalent to the AND operation

A product term is produced by an AND operation with no OR ops involved.i.e.A product term is equal to 1 only if each of the literals in the term is 1.A product term is equal to 0 when one or more of the literals are 0. 00 = 001 = 010 = 011 = 1

Laws & Rules of Boolean AlgebraThe basic laws of Boolean algebra:The commutative lawsThe associative laws The distributive laws

Commutative LawsThe commutative law of addition for two variables is written as: A+B = B+A

The commutative law of multiplication for two variables is written as: AB = BA

ABA+BBAB+AABABBAB+A

Associative LawsThe associative law of addition for 3 variables is written as: A+(B+C) = (A+B)+C

The associative law of multiplication for 3 variables is written as: A(BC) = (AB)CABA+(B+C)CAB(A+B)+CCABA(BC)CAB(AB)CCB+CA+BBCAB

Distributive LawsThe distributive law is written for 3 variables as follows: A(B+C) = AB + ACBCAB+CABCAXXABACX=A(B+C)X=AB+AC

Rules of Boolean Algebra___________________________________________________________A, B, and C can represent a single variable or a combination of variables.

Function Minimization using Boolean Algebra Examples:

(a) a + ab (b) a(a + b) (c) a(a' + b) (d) abc + abc + abc + abc + abc(e) (a+b+c)(a+b+c)(a+b+c)

The other type of questionShow that:1. ab + ab' = a2. (a + b)(a + b') = a

More ExamplesShow that;(a) ab + ab'c = ab + ac(b) (a + b)(a + b' + c) = a + bc

DeMorgans TheoremsDeMorgans theorems provide mathematical verification of: the equivalency of the NAND and negative-OR gatesthe equivalency of the NOR and negative-AND gates.

DeMorgans TheoremsThe complement of two or more ANDed variables is equivalent to the OR of the complements of the individual variables.

The complement of two or more ORed variables is equivalent to the AND of the complements of the individual variables.

NANDNegative-ORNegative-ANDNOR

DeMorgans Theorems (Exercises)Apply DeMorgans theorems to the expressions:

DeMorgans Theorems (Exercises)Apply DeMorgans theorems to the expressions:

Boolean Analysis of Logic CircuitsBoolean algebra provides a concise way to express the operation of a logic circuit formed by a combination of logic gatesso that the output can be determined for various combinations of input values.

Boolean Expression for a Logic CircuitTo derive the Boolean expression for a given logic circuit, begin at the left-most inputs and work toward the final output, writing the expression for each gate.CDBACDB+CDA(B+CD)

Constructing a Truth Table for a Logic CircuitOnce the Boolean expression for a given logic circuit has been determined, a truth table that shows the output for all possible values of the input variables can be developed.Lets take the previous circuit as the example:A(B+CD)There are four variables, hence 16 (24) combinations of values are possible.

Constructing a Truth Table for a Logic CircuitEvaluating the expressionTo evaluate the expression A(B+CD), first find the values of the variables that make the expression equal to 1 (using the rules for Boolean add & mult).In this case, the expression equals 1 only if A=1 and B+CD=1 becauseA(B+CD) = 11 = 1

Constructing a Truth Table for a Logic CircuitEvaluating the expression (cont)Now, determine when B+CD term equals 1.The term B+CD=1 if either B=1 or CD=1 or if both B and CD equal 1 becauseB+CD = 1+0 = 1B+CD = 0+1 = 1B+CD = 1+1 = 1The term CD=1 only if C=1 and D=1

Constructing a Truth Table for a Logic CircuitEvaluating the expression (cont)Summary:A(B+CD)=1 When A=1 and B=1 regardless of the values of C and DWhen A=1 and C=1 and D=1 regardless of the value of BThe expression A(B+CD)=0 for all other value combinations of the variables.

Constructing a Truth Table for a Logic CircuitPutting the results in truth table formatWhen A=1 and B=1 regardless of the values of C and DWhen A=1 and C=1 and D=1 regardless of the value of B

A(B+CD)=1

INPUTSOUTPUTABCDA(B+CD)0000000100100011010001010110011110001001101010111100110111101111

INPUTSOUTPUTABCDA(B+CD)00000001001000110100010101100111100010011010101111001110111110111111

INPUTSOUTPUTABCDA(B+CD)000000010010001101000101011001111000100110101011111001110111110111111

INPUTSOUTPUTABCDA(B+CD)00000000100010000110010000101001100011101000010010101001011111001110111110111111

**********(a) a + ab = a(1+b)=a(b) a(a + b) = a.a +ab=a+ab=a(1+b)=a.(c) a(a' + b) = a. a' +ab=0+ab=ab

(d) abc + abc + abc + abc + abc = c +ab(e) (a+b+c)(a+b+c)(a+b+c)=a+bc

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Solution:1. ab + ab' = a(b+b') = a.1=a2. (a + b)(a + b') = a.a +a.b' +a.b+b.b' = a + a.b' +a.b + 0 = a + a.(b' +b) + 0 = a + a.1 + 0 = a + a = a

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(a) ab + ab'c = a(b + b'c) = a((b+b').(b+c))=a(b+c)=ab+ac

(b) (a + b)(a + b' + c) = (a.a + a.b' + a.c + ab +b.b' +bc) =

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