CHAPTER 3 SETS AND BOOLEAN ALGEBRA PART II – BOOLEAN ALGEBRA & LOGIC CIRCUITS

SETS AND BOOLEAN ALGEBRA 3.5 : EXCURSION: LOGIC CIRCUITS CHAPTER 3 SETS AND BOOLEAN ALGEBRA 3.4 : BOOLEAN ALGEBRA 3.5 : EXCURSION: LOGIC CIRCUITS

3.4 BOOLEAN ALGEBRA INTRODUCTION CONNECTION BETWEEN LOGIC, SETS & BOOLEAN ALGEBRA BOOLEAN OPERATIONS BOOLEAN FUNCTION & EXPRESSION

INTRODUCTION Boolean algebra – provides the operations and the rules for working with the set {0,1} Boolean algebras named after George Boole, who is well known as a pioneer logic. The circuits can be constructed using any basic element that has two different states. Such elements include switches that can be either the on or the off position and optical devices that can either be lit or unlit. We are going to focus on the three operation: -Boolean Complement ; Boolean Sum ; Boolean Product

CONNECTION BETWEEN LOGIC, SETS AND BOOLEAN ALGEBRA PAGE 224

BOOLEAN OPERATIONS AND Also known as Boolean Product. Denoted = dot (.) Has the following values: 1.1=1, 1.0=0, 0.1=0, 0.0=0 OR Also known as Boolean Sum Denoted = sum (+) Has the following values: 1+1=1, 1+0=1, 0+1=1, 0+0=0

BOOLEAN OPERATIONS NOT Also known as complement. Denoted = bar ( ), negation ( ), (‘) It interchanges 0 and 1. Truth Table for AND, OR, NOT operations: a b ab a+b 1

Example 1. Find the value of Answer : 2.

BOOLEAN Function & EXPRESSIONS Let B = {0,1}. The variables x is called a Boolean variable if it assumes values only from B. A function from , the set to B is called Boolean function of degree n . Boolean function can be represented using expressions made up from variables and Boolean operation. The Boolean Expressions in the variables are defined as follows: are Boolean expressions If are Boolean expression then are Boolean expressions.

Example Give a Boolean expression for the Boolean function F(x,y) as defined by the following Answer: is a Boolean function of degree 2 x y F(x,y) 1 Only refer to value 1

BOOLEAN FUNCTIONS & Boolean expressions Each Boolean expression represents a Boolean function. The values of this function are obtained by substituting 0 and 1 for the variables in the expression. Tables that listing the values of function f for all elements of are often called the truth table for f . Example Find the values of the Boolean function represented by 1. 2. 3.

EXERCISE Use a table to express the values of each of these Boolean functions 1. 2. 3. 4. 5. 6. 7.

IDENTITIES OF BOOLEAN ALGEBRA Boolean algebra satisfies many of the same laws as ordinary algebra - addition and multiplication. The following laws are common to both kinds of algebra:

3.5 EXCURSION: LOGIC CIRCUITS LOGIC GATES SIMPLIFYING GATES DUALITY KARNAUGH MAPS CANONICAL FORM

LOGIC GATES A computer or other electric device is made up of a number of circuits. Each circuit can be designed using the rules of Boolean algebra. The basic element of circuits – gates. We will construct combinational circuits using five types of elements: Inverter, OR gate, AND gate, NOR gate and NAND gate

GATES Inverter OR gate AND gate NOR gate NAND gate

LOGIC gates Inverter/NOT: Accept one Boolean variable as input, and produces the complement of this value as output. A A’ 1

LOGIC gates OR gate: The inputs to this gate are the values of two or more Boolean variables, while the output is the sum of their values. A B A+B 1

LOGIC gates AND gate: The inputs to this gate are the values of two or more Boolean variables, while the output is the Boolean product of their values. A B AB 1

LOGIC gates COMBINATION OF GATES NOR Gate This is a NOT-OR gate which is equal to an OR gate followed by a NOT gate. *The outputs of all NOR gates are low if any of the inputs are high. A B 1

LOGIC gates COMBINATION OF GATES NAND Gate This is a NOT-AND gate which is equal to an AND gate followed by a NOT gate. *The outputs of all NAND gates are high if any of the inputs are low. A B 1

EXERCISE Drawing the circuits that produces the output Construct circuits that produce the following outputs: 1. 5. 2. 6. 3. 7. 4. 8.

Given the circuit, find the Boolean function and the output 1 2

3

SIMPLIFYING GATES DUALITY DUALITY METHOD TO SIMPLIFY KARNAUGH MAPS

DUALITY

EXAMPLE

EXAMPLE CAN SIMPLIFY THE BOOLEAN EXPRESSION

CANONICAL FORM Consider a function of three variables x, y, and z. Since each variable may be complemented or uncomplemented, there are different combinations. When combinations are combined with AND, they are called Minterms. When Combinations are combined with OR, they are called Maxterms. Minterm Canonical Form – Standard Products. Maxterm Canonical Form - Standard Sums.

MINTERM & MAXTERM CANONICAL FORM For n Variables there are 2^n Minterms/Maxterms

MINTERM CANONICAL FORM Determine the Set of Minterms for which a function is 1-valued. These are called “Minterms of the Function” Combine all Minterms with a + Operation The sum of minterms that represents the function is called – the sum of products expansion.

MAXTERM CANONICAL FORM Determine the Set of Maxterms for which a function is 0-valued. These are called “Maxterms of the Function” The product of maxterms that represents the function is called – the product-of-sum expansion.

Exercise Find the sum-of-product and the product-of-sum expansion for the function 1. 2. 3. 4. After you get answer for that question, simplify the SOP .

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