CHAPTER 1 SETS, FUNCTIONs, ELEMENTARY LOGIC & BOOLEAN ALGEBRAs BY: MISS FARAH ADIBAH ADNAN IMK

CHAPTER OUTLINE: PART IV 1.4 BOOLEAN ALGEBRAS 1.4.1 INTRODUCTION 1.4.2 BOOLEAN OPERATIONS 1.4.3 BOOLEAN EXPRESSIONS 1.4.4 BOOLEAN FUNCTIONS 1.4.5 DUALITY 1.4.6 IDENTITIES OF BOOLEAN ALGEBRA 1.4.5 LOGIC NETWORKS

1.4.1 INTRODUCTION Boolean algebra – used in design circuits in computers and other electronic devices. The operation of a circuits is defined by a Boolean expression – specify an output for each set of inputs. It deals with values of 0 and 1 / SET {0,1}. Called – bits, binary digits There are 3 operations : AND, OR, and NOT.

1.4.2 BOOLEAN OPERATIONS 1.4.2.1 AND Also known as Boolean Product. Denoted – dot (.) Has the following values: 1.1=1, 1.0=0, 0.1=0, 0.0=0 1.4.2.2 OR Also known as Boolean Product Denoted – sum (+) Has the following values: 1+1=1, 1+0=1, 0+1=1, 0+0=0

1.4.2 BOOLEAN OPERATIONS 1.4.2.3 NOT Also known as complement. Denoted – bar ( ), negation ( ), (‘) It interchanges 0 and 1. Truth Table for AND, OR, NOT operations: X Y XY X+Y X’ 1

Example 1.1 Find the value of Answer :

1.4.3 BOOLEAN EXPRESSIONS The Boolean Expressions in the variables are defined as follows: If are Boolean expressions, then are Boolean expression. Example 1.2: 1) 2)

1.4.4 BOOLEAN FUNCTIONS 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 1.3: Find the values of the Boolean function represented by .

1.4.5 DUALITY The dual of any statement in a Boolean algebra – is the statement obtained by: Interchanging the operations (+) and (.) Interchanging their identity elements 0 and 1 in the original statement Example 1.4: Find the dual of .

1.4.6 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:

1.4.7 LOGIC NETWORKS 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. There are 3 basic types of gates: Inverter, OR gate, and AND gate.

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

1.4.7 LOGIC NETWORKS 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

1.4.7 LOGIC NETWORKS 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

1.4.7 LOGIC NETWORKS 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

1.4.7 LOGIC NETWORKS 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

1.4.8 MINTERM 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.

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

1.4.8 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.

1.4.8 MINTERM CANONICAL FORM The product of maxterms that represents the function is called – the product-of-sum expansion.

1.4.8 MINTERM CANONICAL FORM Example 1.5 Find the sum-of-product expansion for the function .