Boolean Algebra Logic Gates CHAPTER 3 Oxford University

Overview § § § Identify a binary logic variable. Explain the three basic operations of Boolean algebra. Explain some axioms and theorems of Boolean algebra Analyse Boolean expressions and functions and their simplification methods. Explain the different forms of representing a Boolean function. identify logic ‘ true’ and ‘false’ by high and low voltage levels. © Oxford University Press 2013. All rights reserved. 2

Overview § § § Explain gates AND, OR, NOT, NAND, NOR, XOR, and XNOR. Explain the construction of logic gates using electronic devices such as diodes and transistors Use Boolean algebra for describing the function of logic gates Explain how complex logic circuits described by Boolean expressions are constructed using logic gates Construct AND, OR, and NOT gates using NAND and NOR gates © Oxford University Press 2013. All rights reserved. 3

Key Words § § § TRUE: In Boolean algebra true means “ 1”. FALSE: In Boolean algebra false means “ 0”. Boolean algebra: Boolean algebra is the algebra of propositions. It deals with two values, 0 and 1 or true and false. Boolean or logic variable: It is a variable that can be assigned any one of the two values, 0 or 1. Axiom: It is an established statement or proposition. AND: It is an operation in which the output is “ true” only when all the inputs are true. © Oxford University Press 2013. All rights reserved. 4

Key Words § § § OR: It is an operation in which the output is true whenever at least one of the inputs is true. NOT: It is an operation that produces an output which is the complement of the input. NAND: It is an operation in which the output is formed by AND-ing all inputs and then complementing it. NOR: It is an operation in which the output is formed by OR-ing all inputs and then complementing it. Duality: It is the property in which any algebraic equality derived from the axioms of Boolean algebra remains true when the operators OR and AND are interchanged and the identity elements 0 and 1 are interchanged. © Oxford University Press 2013. All rights reserved. 5

Key Words § § § Literal : A literal is a variable or its complement. Example: X, X, Y, Y. Boolean function: A Boolean function is a Boolean variable that has a value, 0 or 1, which gets evaluated from logic computations involving Boolean variables and logic operators like ‘. ’ , ‘ + ’, ‘ — ’. Truth Table: It is a table that depicts the boolean value, 0 or 1, of the output boolean function for different sets of boolean values of the boolean inputs. Term: A term is a collection of boolean variables formed by ANDing or OR-ing , e. g. ABC or (a + c + d). Product term: It is a term formed by AND-ing two or more boolean variables. Sum term: It is a term formed by OR-ing two or more boolean variables. © Oxford University Press 2013. All rights reserved. 6

Key Words § § § Sum of products: It is a function formed with the “ sum “ of product terms. Product of sums: It is a function formed with the “ product “ of sum terms. Canonical form: It is a function formed by min terms or max terms. Min term: It is a special product of literals, in which each input variable appears exactly once. A function with n input variables has 2 n min terms , since each variable can appear complemented or un-complemented. Max term: It is a sum of literals, in which each input variable appears exactly once. A function with n variables has 2 n max terms, because each variable can appear complemented or un-complemented. © Oxford University Press 2013. All rights reserved. 7

Introduction to Boolean Algebra Boolean algebra deals with logic variables, which may either be 1, that is TRUE or 0, that is FALSE. § It uses logic variables and logic operations to develop, manipulate, and simplify logic expressions, following set rules. § Boolean algebra, introduced by George Boole in 1854, differs significantly from conventional algebra. § The rules of Boolean algebra are simple and straightforward, and can be applied to any logical expression. § © Oxford University Press 2013. All rights reserved. 8

Boolean Algebra § The rules of Boolean algebra that define three basic logic operations and some combinations of these, sometimes called axioms, are: 9 © Oxford University Press 2013. All rights reserved.

Theorems § From the previous axioms , we can derive the following theorems. ©

Duality Any algebraic equality derived from the axioms of Boolean algebra remains true when the operators OR and AND are interchanged and the identity elements 0 and 1 are interchanged. § This property is called duality. For example, § ◦ x+1=1 ◦ x. 0 = 0 (dual) Because of the duality principle, for any given theorem its dual may be easily obtained. § The two De Morgan’s theorem are dual of each other. § © Oxford University Press 2013. All rights reserved. 11

Duality © Oxford University Press 2013. All rights reserved. 12

Boolean Expression § A Boolean or logic expression is a logic variable or a number of logic variables involved with one another through the logical operations ‘. ’, ‘+’, and ‘–’. For logic variables A and B, the following are some examples of Boolean expressions: 13 © Oxford University Press 2013. All rights reserved.

Example 14 © Oxford University Press 2013. All rights reserved.

Boolean Function & Truth Tables A Boolean function of one or more logic variables, also known as Boolean variable, is a binary variable, the value of which depends on the values of these logic variables. § For example, independent Boolean variables A and B may have arbitrarily chosen values while the Boolean function f (A, B) has values that depend on the values of A and B, hence: § § The truth table for this function is shown in the Table : 15 © Oxford University Press 2013. All rights reserved.

Constructing Truth Tables from Boolean Expressions § A Boolean function can be built from the value of a given truth table. Considering previous Table, the value of f(A, B) is 1 when ◦ ◦ The above conditions may also be written as ◦ These conditions may further be rewritten as the following logical products: ◦ ◦ Terms: A combination of logic variables forming a group in a Boolean function is called a term. ◦ Literals: Each complemented or uncomplemented variable in a term is called a literal. © Oxford University Press 2013. All rights reserved. 16

Canonical & Standard Form § In a Boolean function, if all terms are written as AND combinations of the Boolean variables, there are 2 n such AND ‘terms’ for n variables. ◦ These AND terms are called min terms. ◦ Min terms are designated as m 0, m 1, . . . mn, etc. , where the subscripts represent the decimal values obtained from the equivalent binary value of the combined variables. © Oxford University Press 2013. All rights reserved. 17

Numerical Representation : Boolean Function in Canonical Form § A Boolean function, with the canonical sum of product terms, can be expressed in a compact form by listing the decimal value corresponding to the min term for which the function value is 1. ◦ As an example, the truth table of a three-variable function is shown below. Three variables, each of which can take the values 0 or 1, yield eight possible combinations of values for which the function may be true. 18 © Oxford University Press 2013. All rights reserved.

Logic Gates § The Boolean functions or expressions can be realized by using electronic gates. ◦ It must be understood that the logic `1 ` and logic `0’, which are fed as input to the gates, are represented by two distinct voltage levels. ◦ Even the output, which is either logic `1’ or `0’, is represented by distinct voltage levels. ◦ There are three fundamental logical operations from which all other Boolean functions, no matter how complex, can be derived. ◦ These operations are implemented by three basic gates: AND, OR, and NOT. ◦ Four other gates NAND, NOR, XOR, and XNOR, which are derived gates, are also used to construct logic functions. © Oxford University Press 2013. All rights reserved. 19

AND Gate The AND gate is an electronic circuit that has two or more inputs and only one output. § It gives a HIGH output (1) only if all its inputs are HIGH. If A and B are logic inputs to a two input AND gate, then output Y is equal to A. B. § The dot (. ) indicates an AND operation. The AND gate is also called an all or nothing gate. § The truth table for the AND gate is given in Table : § © Oxford University Press 2013. All rights reserved. 20

AND Gate § This dot is usually omitted, as shown in the output in

OR Gate The OR gate is an electronic circuit that has two or more inputs and only one output. § It gives a HIGH output if one or more of its inputs are HIGH. For a two-input OR gate, where A and B are the logic inputs, the output Y is equal to A + B. § A plus (+) indicates an OR operation. § The truth table for a two input OR gate is given in Table: § 22 © Oxford University Press 2013. All rights reserved.

Two Input OR Gate © Oxford University Press 2013. All rights reserved. 23

NOT gate or INVERTER The inverter is a little different from AND and OR gates as it has only one input and one output. Whatever logic state is applied to the input, the opposite state will appear at the output. § The NOT function is denoted by a horizontal bar over the value to be inverted, as shown in the Fig. § In some cases, a prime symbol (`) may also be used for this purpose: 0` is 1 and 1` is 0. § 24 © Oxford University Press 2013. All rights reserved.

NOT gate or INVERTER § In the inverter symbol shown in Fig. 25 ©

NAND Gate § The NAND gate implements the NAND function, which means NOT -AND. ◦ The inputs are AND & then NOT to get a single output. ◦ The output of NAND gate is HIGH if any or all of the inputs are LOW. ◦ When all inputs are HIGH, the output is LOW. Below table depicts the truth table for a two-input NAND gate. 26 © Oxford University Press 2013. All rights reserved.

NAND Gate § In Fig. , the circle at the output of the NAND

NOR Gate § The NOR gate is an OR gate with inverted output. ◦ Whereas the OR gate allows the output to be HIGH (logic 1) if any one or more of its inputs are HIGH, the NOR gate inverts this and forces the output to logic 0 when any input is HIGH, i. e. , the output of a NOR gate is LOW if any of the inputs are HIGH. ◦ The output is HIGH when all inputs are LOW. ◦ The NOR function uses the plus sign (+) operator with the output represented by an expression with an over bar to indicate the OR inversion. ◦ The truth table of a two-input NOR gate is given in Table. 28 © Oxford University Press 2013. All rights reserved.

NOR Gate § The NOR function can also be performed by a bubbled AND

XOR Gate § The Exclusive-OR or XOR gate is a two-input circuit that will give a HIGH output if either, but not both, of the inputs are HIGH. § The truth table of XOR gate is given in Table. © Oxford University Press 2013. All rights reserved. 30

XOR Gate § An encircled plus operation. sign is used to show the XOR

XNOR Gate § The Exclusive-NOR gate is a XOR gate followed by a NOT gate. XNOR gate is a two-input and one-output logic gate circuit. § In the gate, the output is HIGH if both inputs are either LOW or HIGH. The logic symbol for a XNOR is shown in Fig. 32 © Oxford University Press 2013. All rights reserved.

Describing Logic Circuits Algebraically Any logic circuit, no matter how complex, may be completely described using the Boolean operations because the OR, AND, and NOT gates are the basic building blocks of digital systems. § The algebraic expression that relates the logic output of a logic circuit with the binary inputs of the logic circuit, is called a Boolean expression. § © Oxford University Press 2013. All rights reserved. 33

Describing Logic Circuits Algebraically If an expression contains both AND & OR operations, the AND operations are performed first. § For example, in Y = AB + C, AB is performed first, unless there are parentheses in the expression, in which case the operation inside the parentheses is performed first. § That is, in Y = (A + B) + C, A + B is performed first. § 34 © Oxford University Press 2013. All rights reserved.

Realization of Logic Circuits from Boolean Expression If the operation of a circuit is defined by a Boolean expression, a logic-circuit diagram can be developed directly from the expression. § Suppose a circuit has to be constructed whose output is Y = AC + B`C + A`BC. § This implies that a three-input OR gate is required with inputs that are equal to AC, B`C and A`BC respectively. § 35 © Oxford University Press 2013. All rights reserved.

Universality : NAND & NOR Gates NAND and NOR are called universal gates since the AND, OR, and NOT gates can be constructed with either of them. ◦ It is possible to implement any logic expression using only NAND gates. ◦ This is because NAND gates, in the proper combination, can be used to perform each of the Boolean operations OR, AND, and NOT. ◦ Figure shows how NAND gates are used to implement AND, OR, and NOT operations. 36 © Oxford University Press 2013. All rights reserved.