Boolean Algebra and Logic Gates


 Elizabeth Merritt
 2 years ago
 Views:
Transcription
1 Boolean Algebra and Logic Gates Binary logic is used in all of today's digital computers and devices Cost of the circuits is an important factor Finding simpler and cheaper but equivalent circuits can reduce the overall cost Mathematical methods that simplify circuits rely primarily on Boolean algebra Explore basic vocabulary and a brief foundation in Boolean algebra Optimize simple circuits Understand software tools which optimize very large circuits 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 1
2 Basic Definitions A set of elements is any collection of objects usually having a common property If S is a set x S means that x is a member of the set S y S means that y is not a member of the set S A = {1, 2, 3, 4} indicates that the elements of set A are the numbers 1, 2, 3, and 4 A binary operator defined on a set S of elements is a rule that assigns, to each pair of elements from S, a unique element from S a * b = c : * is a binary operator if it specifies a rule for finding c from the pair (a, b) and also if a, b, c S * is not a binary operator if a, b S, and if c S 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 2
3 Basic Definitions Postulates form the basic assumptions to deduce rules, theorems, and properties 1. Closure: A set S is closed with respect to a binary operator if, for every pair of elements of S, the binary operator specifies a rule for obtaining a unique element of S 2. Associative law: A binary operator * on a set S is said to be associative whenever: (x * y) * z = x * (y * z) for all x, y, z, S 3. Commutative law: A binary operator * on a set S is said to be commutative whenever: x * y = y * x for all x, y S 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 3
4 Basic Definitions 4. Identity element: A set S is said to have an identity element with respect to a binary operation * on S if there exists an element e S with the property that: e * x = x * e = x for every x S 5. Inverse: A set S having the identity element e with respect to a binary operator * is said to have an inverse whenever, for every x S, there exists an element y S such that: x * y = e 6. Distributive law: If * and are two binary operators on a set S, * is said to be distributive over whenever: x * (y z) = (x * y) (x * z) 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 4
5 Field of Real Numbers The binary operator + defines addition The additive identity is 0 The additive inverse defines subtraction The binary operator defines multiplication The multiplicative identity is 1 For a 0, the multiplicative inverse of a = 1/a defines division: a 1 / a = 1 The only distributive law applicable is that of over + : a (b + c) = (a b) +(a c) 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 5
6 Boolean Algebra Boolean algebra (George Boole) is an algebraic structure defined by a set of elements, B, together with two binary operators, + and, satisfying the following postulates (E.V. Huntington): 1a. The structure is closed with respect to the operator + 1b. The structure is closed with respect to the operator 2a. The element 0 is an identity element with respect to +: x + 0 = 0 + x = x 2b. The element 1 is an identity element with respect to : x 1 = 1 x = x 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 6
7 Boolean Algebra 3a. The structure is commutative with respect to +: x + y = y + x 3b. The structure is commutative with respect to : x y = y x 4a. The operator is distributive over +: x (y + z) = (x y) + (x z) 4b. The operator + is distributive over : x + (y z) = (x + y) (x + z) 5. For every element x B, there exists an element x' B (called the complement of x) such that: (a) x + x' = 1 and (b) x x' = 0 6. There exists at least two elements x, y B such that x y 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 7
8 Differences between Boolean and Ordinary (Real Field) Algebra Ordinary (Real Field) Algebra Huntington postulates do not include the associative law but can be derived for both operators The distributive law of + over, x + (y z) = (x + y) (x + z), is valid for Boolean algebra, but not for ordinary algebra Boolean algebra does not have additive or multiplicative inverses; no subtraction or division operations Postulate 5 defines an operator called the complement that is not available in ordinary algebra Ordinary algebra deals with the real numbers, which constitute an infinite set of elements Boolean algebra deals with the as yet undefined set of elements, B, but in the twovalued Boolean algebra (what we will be using), B is defined as a set with only two elements, 0 and Roberto Muscedere Images 2013 Pearson Education Inc. 8
9 Differences between Boolean and Ordinary (Real Field) Algebra The choice of the symbols + and is intentional Operations are similar but not identical Know the difference between elements (0 and 1) and variables (ie. x and y) which are just symbols To have a Boolean algebra, we must show: the elements of the set B the rules of operation for the two binary operators the set of elements, B, together with the two operators, satisfy the six Huntington postulates Will use Twovalued Boolean Algebra 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 9
10 TwoValued Boolean Algebra Defined on a set of two elements, B = {0, 1}, with rules for the two binary operators + and Same as AND, OR, and NOT operations Must show that Huntington postulates are valid for the set B = {0, 1}, with the two binary operators + and 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 10
11 TwoValued Boolean Algebra 1. That the structure is closed with respect to the two operators since the result of each operation is either 1 or 0 and 1, 0 B 2. From the tables: (a) = = = 1 (b) 1 1 = = 0 1 = 0 This establishes the two identity elements, 0 for + and 1 for 3. The commutative laws apply from the symmetry of the binary operator tables 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 11
12 TwoValued Boolean Algebra 4(a). The distributive law x (y + z) = (x y) + (x z) can be shown to hold by forming a truth table of all possible values of x, y, and z 4(b). The distributive law of + over can be shown to hold by means of a truth table just as in (a) 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 12
13 TwoValued Boolean Algebra 5. From the complement table: (a) x + x' = 1, since: 0 + 0' = = 1 and 1 + 1' = = 1 (b) x x' = 0, since: 0 0' = 0 1 = 0 and 1 1' = 1 0 = 0 6. Postulate 6 is satisfied since twovalued Boolean algebra has two elements, 1 and 0, with 1 0 Twovalued Boolean algebra has been defined and is equivalent to binary logic shown earlier 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 13
14 Duality Principle of Boolean Algebra Duality principle states that every algebraic expression deducible from the postulates of Boolean algebra remain valid if the operators and identity elements are interchanged Interchange OR and AND operators and replace 1's by 0's and 0's by 1's 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 14
15 Basic Theorems and Postulates of Boolean Algebra Basic relationships in Boolean algebra Listed in pairs; dual of each other Proofs in the text book 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 15
16 Operator Precedence for Boolean Algebra In the following order: Parentheses, NOT, AND, and OR Like BEDMAS, but E is complement, DM is AND, and AS is OR Example: (x + y)': OR is done first, then complement x'y': Each variable is complemented, then ANDed 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 16
17 Boolean Functions Boolean algebra is an algebra that deals with binary variables and logic operations A Boolean function described by an algebraic expression consists of binary variables, the constants 0 and 1, and the logic operation symbols For a given value of the binary variables, the function can be equal to either 1 or 0 Example: F 1 = x + y'z F 1 is 1 when x = 1 or if both y' and z are equal to Roberto Muscedere Images 2013 Pearson Education Inc. 17
18 Boolean Functions A Boolean function can be represented in a truth table F 1 = x + y'z Number of rows is 2 n n is the number of variables in the function Binary combinations are obtained from the binary numbers by counting from 0 through 2 n Roberto Muscedere Images 2013 Pearson Education Inc. 18
19 Boolean Functions A Boolean function can be transformed from an algebraic expression into a circuit diagram composed of logic gates connected in a particular structure Rather than listing each combination of inputs and outputs, it indicates how to compute the logic value of each output from the logic values of the inputs Example: F 1 = x + y'z 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 19
20 Boolean Functions A Boolean function can only be represented in a truth table in one way In algebraic form it can be expressed in a variety of ways all of which have equivalent logic Expression used will dictate the interconnection of gates Conversely the interconnection of gates will dictate the logic expression By manipulating a Boolean expression according to the rules of Boolean algebra, it is sometimes possible to obtain a simpler expression for the same function reduce the number of gates in the circuit and the number of inputs to the gate can significantly reduce the cost of a circuit 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 20
21 Boolean Functions F 2 = x'y'z + x'yz + xy' 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 21
22 Boolean Functions F 2 = x'y'z + x'yz + xy' = x'z(y' + y) + xy' = x'z + xy' Verify with truth table 1st circuit had 3 terms, 8 literals 2nd circuit has 2 terms, 4 literals 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 22
23 Algebraic Manipulation 1 and 2 are the duals of each other 3 from postulate 4b: x + yz = (x + y)(x+ z) 4 and 5 are together known as the consensus theorem 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 23
24 Complement of a Function The generalized form of DeMorgan's theorems states that the complement of a function is obtained by interchanging AND and OR operators and complementing each literal: (A + B + C + D F)' = A'B'C'D'...F' (ABCD...F)' = A' + B' + C' + D' F' 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 24
25 Complement of a Function Find the complement of: F 1 = x'yz' + x'y'z and F 2 = x(y'z' + yz): (A + B + C + D F)' = A'B'C'D'...F' (ABCD...F)' = A' + B' + C' + D' F' 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 25
26 Complement of a Function A simpler procedure for deriving the complement of a function is to take the dual of the function and complement each literal: F 1 = x'yz' + x'y'z The dual of F 1 :(x' + y + z')(x' + y' + z) Complement each literal: (x + y' + z)(x+ y + z') = F' 1 F 2 = x(y'z' + yz) The dual of F 2 : x + (y' + z')(y+ z) Complement each literal: x' + (y + z)(y' + z') = x' + yy' + yz' + y'z + zz' = F' Roberto Muscedere Images 2013 Pearson Education Inc. 26
27 Minterms A binary variable may appear in normal (x) or complement form (x') Two binary variables x and y combined with an AND operation There are four possible combinations: x'y', x'y, xy', and xy Each of these four AND terms is called a minterm, or a standard product n variables can be combined to form 2 n minterms Each minterm is obtained from an AND term of the n variables each variable being primed if the corresponding bit of the binary number (0 to 2 n 1) is a 0 and unprimed if a 1 A symbol for each minterm is of the form m j, where the subscript j denotes the decimal equivalent of the binary number of the minterm designated 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 27
28 Maxterms In a similar fashion, n variables forming an OR term, with each variable being primed or unprimed, provide 2 n maxterms, or a standard sum Each maxterm is obtained from an OR term of the n variables each variable being unprimed if the corresponding bit of the binary number (0 to 2 n 1) is a 0 and primed if a 1 A symbol for each maxterm is of the form M j, where the subscript j denotes the decimal equivalent of the binary number of the maxterm designated 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 28
29 Minterms and Maxterms Each minterm is the complement of its corresponding maxterm and vice versa 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 29
30 Minterms and Maxterms A Boolean function can be expressed algebraically from a truth table by forming a minterm for each combination of the variables that produces a 1 and then taking the OR of all those terms Sum of minterms Example: f 1 = x'y'z + xy'z' + xyz = m 1 + m 4 + m 7 f 2 = x'yz + xy'z + xyz' + xyz = m 3 + m 5 + m 6 + m Roberto Muscedere Images 2013 Pearson Education Inc. 30
31 Minterms and Maxterms Complement of a Boolean function may be produced by creating a sum of minterms from the truth table outputs which are 0 f ' 1 = x'y'z' + x'yz' + x'yz + xy'z + xyz' Take complement: f 1 = (x + y + z)(x + y' + z)(x' + y + z')(x' + y' + z) = M 0 M 2 M 3 M 5 M 6 Generate f 2 from 0 outputs on table: f 2 = (x + y + z)(x + y + z')(x + y' + z)(x' + y + z) = M 0 M 1 M 2 M 4 Any Boolean function can be expressed as a product of maxterms Boolean functions expressed as a sum of minterms or product of maxterms are said to be in canonical form 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 31
32 Sum of Minterms The minterms whose sum defines the Boolean function are those which give the 1's of the function in a truth table If the function is not in this form (one or more variables missing from each minterm) it can be ANDed with an expression such as (x + x') F = A + B'C A= A(B + B') = AB + AB' A= AB(C + C') + AB'(C + C') = ABC + ABC' + AB'C + AB'C' B'C = B'C(A + A') = AB'C + A'B'C F = A + B'C = ABC + ABC' + AB'C + AB'C' + AB'C + A'B'C Remove duplicate minterms and rearrange F = A'B'C + AB'C' + AB'C + ABC' + ABC = m 1 + m 4 + m 5 + m 6 + m Roberto Muscedere Images 2013 Pearson Education Inc. 32
33 Sum of Minterms Sometimes convenient to express the function in a briefer notation: F(A, B, C) = (1, 4, 5, 6, 7) Alternative procedure: Read minterms (1 outputs) directly from truth table 1, 4, 5, 6, Roberto Muscedere Images 2013 Pearson Education Inc. 33
34 Product of Maxterms Can also express functions as a product of maxterms If the function is not in this form, use distributive law: x + yz = (x + y)(x + z) Missing variable(s), x for example, term is ORed with xx' F= xy+ x'z= (xy + x')(xy + z) = (x + x')(y+ x')(x+ z)(y + z) = (x' + y)(x+ z)(y+ z) OR each term with missing variables: x' + y = x' + y + zz' = (x' + y + z)(x' + y + z') x + z = x + z + yy' = (x + y + z)(x + y' + z) y + z = y + z + xx' = (x + y + z)(x' + y + z) 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 34
35 Product of Maxterms Remove duplicate maxterms and rearrange F= (x + y + z)(x + y' + z)(x' + y + z)(x' + y + z') = M 0 M 2 M 4 M 5 Sometimes convenient to express the function in a briefer notation: F(x, y, z) = (0, 2, 4, 5) Alternative procedure: Read maxterms (0 outputs) directly from truth table 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 35
36 Conversion between Canonical Forms The complement of a function expressed as the sum of minterms equals the sum of minterms missing from the original function F(A, B, C) = (1, 4, 5, 6, 7) F'(A, B, C) = (0, 2, 3) = m 0 + m 2 + m 3 F = (m 0 + m 2 + m 3 )' = m' 0 m' 2 m' 3 = M 0 M 2 M 3 = (0, 2, 3) The maxterm with subscript j is a complement of the minterm with the same subscript j and vice versa m' j = M j 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 36
37 Conversion between Canonical Example: F = xy + x'z Forms Derive truth table Minterms have 1 output F(x, y, z) = (1, 3, 6, 7) Maxterms have 0 output F(x, y, z) = (0, 2, 4, 5) 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 37
38 Standard Forms The two canonical forms of Boolean algebra are basic forms generated from a truth table Very seldom used since each minterm or maxterm must contain all the variables which generates the most literals Another way is the standard form Terms that form the function may contain one, two, or any number of literals Two types: Sum of products Products of sums 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 38
39 Sum of Products Boolean expression containing AND terms (product terms) with one or more literals each The sum denotes the ORing of these terms F 1 = y' + xy + x'yz' Input complements are assumed to be available 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 39
40 Product of Sums Boolean expression containing OR terms (sum terms) with one or more literals each The product denotes the ANDing of these terms F 2 = x(y' + z)(x' + y + z') 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 40
41 NonStandard Form May be expressed in a nonstandard form Neither a sum of products or product of sums F 3 = AB + C(D + E) (a) requires two AND gates, two OR gates and is 3 levels Using distributive law to remove the parentheses: F 3 = AB + C(D + E) = AB + CD + CE (b) requires three AND gates, one OR gate, and is 2 levels 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 41
42 Other Logic Operations 2 2n functions for n binary variables When n = 2, there are 16 possible Boolean functions AND and OR are only 2 of the 16 functions Three categories: 1. Two functions that produce a constant 0 or 1 2. Four functions with unary operations: complement and transfer 3. Ten functions with binary operators that define eight different operations: AND, OR, NAND, NOR, exclusiveor, equivalence, inhibition, and implication 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 42
43 Other Logic Operations 16 Functions of Two Variables 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 43
44 Other Logic Operations 16 Functions of Two Variables 16 Functions of Two Variables Typically don't use constants and transfers as they can be optimized out of a function Inhibition and implication are seldom used in computer logic AND, OR and complement have been used in Boolean algebra NOR is the complement of OR (notor) NAND is the complement of AND (notand) ExclusiveOR (XOR) is 1 when x and y differ in value Equivalence is the complement of XOR (XNOR), is 1 when x and y are the same 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 44
45 Digital Logic Gates  Inverter Small circle (referred to as a bubble) designates the logic complement Requires 2 transistors Triangle symbol designates a buffer circuit or transfer function Used for power amplification of the signal and is equivalent to two inverters connected in cascade 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 45
46 Digital Logic Gates  OR/NOR OR requires 6 transistors NOR is complement of OR; bubble at output Requires 4 transistors; desirable 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 46
47 Digital Logic Gates  AND/NAND AND requires 6 transistors NAND is complement of AND; bubble at output Requires 4 transistors; faster than NOR; more desirable 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 47
48 Digital Logic Gates  XOR/XNOR XOR requires 12 transistors XNOR is complement of XOR; bubble at output Requires 12 transistors; function is desirable 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 48
49 Extension to Multiple Inputs A gate can be extended to have multiple inputs if the binary operation it represents is commutative and associative AND, OR, XOR, and XNOR can be easily extended NAND and NOR functions are commutative, but not associative For now, we define the multiple NOR (or NAND) gate as a complemented OR (or AND) gate: x y z = (x + y + z)' x y z = (xyz)' 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 49
50 Positive and Negative Logic Highlevel H representing logic 1 defines a positive logic system Lowlevel L representing logic 1 defines a negative logic system It is the designers choice 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 50
51 Positive and Negative Logic To switch between positive and negative logic: Invert the input and outputs (switch 0's and 1's) This produces the dual of a function All AND operations are converted to OR operations and vice versa Use polarity indicator (triangular bubble) 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 51
52 Integrated Circuits An integrated circuit (IC) is fabricated on a die of a silicon semiconductor crystal, called a chip, containing the electronic components for constructing digital gates We will not cover the process for creating chips The various gates are interconnected inside the chip to form the required circuit The chip is mounted in a ceramic or plastic container, and connections are welded to external pins Number of pins may range from 8 to several thousand Data sheets contain descriptions and information about the ICs ICs usually have some identification printed on them 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 52
53 Levels of Integration Smallscale integration (SSI): Contain less than 10 independent gates which are connected directly to the pins in the package. Mediumscale integration (MSI): Contain 10 to 1000 gates that perform specific elementary digital operations Decoders, adders, multiplexers, registers, and counters Largescale integration (LSI): Contain thousands of gates Processors, memory chips, and programmable logic devices Very largescale integration (VLSI): Contain millions of gates Large memory arrays and complex microcomputers 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 53
54 Digital Logic Families Each logic family has its own basic electronic circuit upon which more complex digital circuits and components are developed Many different logic families: 1. TTL transistortransistor logic In use for 50 years 2. ECL emittercoupled logic Highspeed operation 3. MOS metaloxide semiconductor High component density 4. CMOS complementary MOS Low power; dominant family 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 54
55 Digital Logic Families Important parameters distinguishing logic families: 1. Fanout: Number of standard loads that the output can drive without impairing its normal operation 2. Fanin: Number of inputs available 3. Power dissipation: Power consumed that must be available from the power supply 4. Propagation delay: Average transition delay time for a signal to propagate from input to output 5. Noise margin: Maximum external noise voltage added to an input signal that does not cause an undesirable change in the output 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 55
56 ComputerAided Design (CAD) VLSI circuits contain millions of transistors Systems are very complex Use CAD to aid in development and verification Electronic Design Automation (EDA) are specific to the VLSI area Cover all necessary steps to design and build a circuit Various targets: 1. Applicationspecific integrated circuit (ASIC) 2. Fieldprogrammable gate array (FPGA) 3. Programmable logic device (PLD) 4. Fullcustom IC Choice based on market and unit costs 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 56
57 Hardware Description Languages (HDLs) Resembles a computer programming language Oriented to describing digital hardware functionality Two leading types: Verilog and VHDL Can be abstract (no reference to specific hardware) allows designer to devote attention to higher level details HDLbased models are simulated to check and verify its functionality before fabrication EDA tools have been developed to synthesize the logic described by an HDL model 2018 Roberto Muscedere Images 2013 Pearson Education Inc. 57
Chapter 2. Boolean Algebra and Logic Gates
Chapter 2. Boolean Algebra and Logic Gates Tong In Oh 1 Basic Definitions 2 3 2.3 Axiomatic Definition of Boolean Algebra Boolean algebra: Algebraic structure defined by a set of elements, B, together
More informationLecture (05) Boolean Algebra and Logic Gates
Lecture (05) Boolean Algebra and Logic Gates By: Dr. Ahmed ElShafee ١ Minterms and Maxterms consider two binary variables x and y combined with an AND operation. Since eachv ariable may appear in either
More informationBoolean Algebra. BME208 Logic Circuits Yalçın İŞLER
Boolean Algebra BME28 Logic Circuits Yalçın İŞLER islerya@yahoo.com http://me.islerya.com 5 Boolean Algebra /2 A set of elements B There exist at least two elements x, y B s. t. x y Binary operators: +
More informationChap2 Boolean Algebra
Chap2 Boolean Algebra Contents: My name Outline: My position, contact Basic information theorem and postulate of Boolean Algebra. or project description Boolean Algebra. Canonical and Standard form. Digital
More informationGate Level Minimization Map Method
Gate Level Minimization Map Method Complexity of hardware implementation is directly related to the complexity of the algebraic expression Truth table representation of a function is unique Algebraically
More information2.1 Binary Logic and Gates
1 EED2003 Digital Design Presentation 2: Boolean Algebra Asst. Prof.Dr. Ahmet ÖZKURT Asst. Prof.Dr Hakkı T. YALAZAN Based on the Lecture Notes by Jaeyoung Choi choi@comp.ssu.ac.kr Fall 2000 2.1 Binary
More informationChapter 2 Boolean algebra and Logic Gates
Chapter 2 Boolean algebra and Logic Gates 2. Introduction In working with logic relations in digital form, we need a set of rules for symbolic manipulation which will enable us to simplify complex expressions
More informationLecture (04) Boolean Algebra and Logic Gates
Lecture (4) Boolean Algebra and Logic Gates By: Dr. Ahmed ElShafee ١ Dr. Ahmed ElShafee, ACU : Spring 26, Logic Design Boolean algebra properties basic assumptions and properties: Closure law A set S is
More informationLecture (04) Boolean Algebra and Logic Gates By: Dr. Ahmed ElShafee
Lecture (4) Boolean Algebra and Logic Gates By: Dr. Ahmed ElShafee Boolean algebra properties basic assumptions and properties: Closure law A set S is closed with respect to a binary operator, for every
More informationGet Free notes at ModuleI One s Complement: Complement all the bits.i.e. makes all 1s as 0s and all 0s as 1s Two s Complement: One s complement+1 SIGNED BINARY NUMBERS Positive integers (including zero)
More informationCombinational Logic & Circuits
WeekI Combinational Logic & Circuits Spring' 232  Logic Design Page Overview Binary logic operations and gates Switching algebra Algebraic Minimization Standard forms Karnaugh Map Minimization Other
More information2.6 BOOLEAN FUNCTIONS
2.6 BOOLEAN FUNCTIONS Binary variables have two values, either 0 or 1. A Boolean function is an expression formed with binary variables, the two binary operators AND and OR, one unary operator NOT, parentheses
More informationIT 201 Digital System Design Module II Notes
IT 201 Digital System Design Module II Notes BOOLEAN OPERATIONS AND EXPRESSIONS Variable, complement, and literal are terms used in Boolean algebra. A variable is a symbol used to represent a logical quantity.
More informationChapter 2. Boolean Expressions:
Chapter 2 Boolean Expressions: A Boolean expression or a function is an expression which consists of binary variables joined by the Boolean connectives AND and OR along with NOT operation. Any Boolean
More informationBinary logic. Dr.AbuArqoub
Binary logic Binary logic deals with variables like (a, b, c,, x, y) that take on two discrete values (, ) and with operations that assume logic meaning ( AND, OR, NOT) Truth table is a table of all possible
More informationMenu. Algebraic Simplification  Boolean Algebra EEL3701 EEL3701. MSOP, MPOS, Simplification
Menu Minterms & Maxterms SOP & POS MSOP & MPOS Simplification using the theorems/laws/axioms Look into my... 1 Definitions (Review) Algebraic Simplification  Boolean Algebra Minterms (written as m i ):
More informationCS8803: Advanced Digital Design for Embedded Hardware
CS883: Advanced Digital Design for Embedded Hardware Lecture 2: Boolean Algebra, Gate Network, and Combinational Blocks Instructor: Sung Kyu Lim (limsk@ece.gatech.edu) Website: http://users.ece.gatech.edu/limsk/course/cs883
More informationGateLevel Minimization. BME208 Logic Circuits Yalçın İŞLER
GateLevel Minimization BME28 Logic Circuits Yalçın İŞLER islerya@yahoo.com http://me.islerya.com Complexity of Digital Circuits Directly related to the complexity of the algebraic expression we use to
More informationCombinational Circuits
Combinational Circuits Combinational circuit consists of an interconnection of logic gates They react to their inputs and produce their outputs by transforming binary information n input binary variables
More informationLecture 4: Implementation AND, OR, NOT Gates and Complement
EE210: Switching Systems Lecture 4: Implementation AND, OR, NOT Gates and Complement Prof. YingLi Tian Feb. 13, 2018 Department of Electrical Engineering The City College of New York The City University
More informationUnitIV Boolean Algebra
UnitIV Boolean Algebra Boolean Algebra Chapter: 08 Truth table: Truth table is a table, which represents all the possible values of logical variables/statements along with all the possible results of
More informationObjectives: 1 Bolean Algebra. Eng. Ayman Metwali
Objectives: Chapter 3 : 1 Boolean Algebra Boolean Expressions Boolean Identities Simplification of Boolean Expressions Complements Representing Boolean Functions 2 Logic gates 3 Digital Components 4
More informationCombinational Logic Circuits
Chapter 2 Combinational Logic Circuits J.J. Shann (Slightly trimmed by C.P. Chung) Chapter Overview 21 Binary Logic and Gates 22 Boolean Algebra 23 Standard Forms 24 TwoLevel Circuit Optimization
More informationDIGITAL CIRCUIT LOGIC UNIT 7: MULTILEVEL GATE CIRCUITS NAND AND NOR GATES
DIGITAL CIRCUIT LOGIC UNIT 7: MULTILEVEL GATE CIRCUITS NAND AND NOR GATES 1 iclicker Question 13 Considering the KMap, f can be simplified as (2 minutes): A) f = b c + a b c B) f = ab d + a b d AB CD
More informationGate Level Minimization
Gate Level Minimization By Dr. M. Hebaishy Digital Logic Design Ch Simplifying Boolean Equations Example : Y = AB + AB Example 2: = B (A + A) T8 = B () T5 = B T Y = A(AB + ABC) = A (AB ( + C ) ) T8 =
More informationChapter 3. GateLevel Minimization. Outlines
Chapter 3 GateLevel Minimization Introduction The Map Method FourVariable Map FiveVariable Map Outlines Product of Sums Simplification Don tcare Conditions NAND and NOR Implementation Other TwoLevel
More informationCONTENTS CHAPTER 1: NUMBER SYSTEM. Foreword...(vii) Preface... (ix) Acknowledgement... (xi) About the Author...(xxiii)
CONTENTS Foreword...(vii) Preface... (ix) Acknowledgement... (xi) About the Author...(xxiii) CHAPTER 1: NUMBER SYSTEM 1.1 Digital Electronics... 1 1.1.1 Introduction... 1 1.1.2 Advantages of Digital Systems...
More informationVariable, Complement, and Literal are terms used in Boolean Algebra.
We have met gate logic and combination of gates. Another way of representing gate logic is through Boolean algebra, a way of algebraically representing logic gates. You should have already covered the
More information1. Mark the correct statement(s)
1. Mark the correct statement(s) 1.1 A theorem in Boolean algebra: a) Can easily be proved by e.g. logic induction b) Is a logical statement that is assumed to be true, c) Can be contradicted by another
More informationSoftware Engineering 2DA4. Slides 2: Introduction to Logic Circuits
Software Engineering 2DA4 Slides 2: Introduction to Logic Circuits Dr. Ryan Leduc Department of Computing and Software McMaster University Material based on S. Brown and Z. Vranesic, Fundamentals of Digital
More informationChapter 3. Boolean Algebra and Digital Logic
Chapter 3 Boolean Algebra and Digital Logic Chapter 3 Objectives Understand the relationship between Boolean logic and digital computer circuits. Learn how to design simple logic circuits. Understand how
More informationBoolean algebra. June 17, Howard Huang 1
Boolean algebra Yesterday we talked about how analog voltages can represent the logical values true and false. We introduced the basic Boolean operations AND, OR and NOT, which can be implemented in hardware
More informationGateLevel Minimization
GateLevel Minimization ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2017 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outlines The Map Method
More informationX Y Z F=X+Y+Z
This circuit is used to obtain the compliment of a value. If X = 0, then X = 1. The truth table for NOT gate is : X X 0 1 1 0 2. OR gate : The OR gate has two or more input signals but only one output
More informationExperiment 3: Logic Simplification
Module: Logic Design Name:... University no:.. Group no:. Lab Partner Name: Mr. Mohamed ElSaied Experiment : Logic Simplification Objective: How to implement and verify the operation of the logical functions
More informationSWITCHING THEORY AND LOGIC CIRCUITS
SWITCHING THEORY AND LOGIC CIRCUITS COURSE OBJECTIVES. To understand the concepts and techniques associated with the number systems and codes 2. To understand the simplification methods (Boolean algebra
More informationAssignment (36) Boolean Algebra and Logic Simplification  General Questions
Assignment (36) Boolean Algebra and Logic Simplification  General Questions 1. Convert the following SOP expression to an equivalent POS expression. 2. Determine the values of A, B, C, and D that make
More informationAnnouncements. Chapter 2  Part 1 1
Announcements If you haven t shown the grader your proof of prerequisite, please do so by 11:59 pm on 09/05/2018 (Wednesday). I will drop students that do not show us the prerequisite proof after this
More informationELCT201: DIGITAL LOGIC DESIGN
ELCT201: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 3 Following the slides of Dr. Ahmed H. Madian محرم 1439 ه Winter
More informationUNIT4 BOOLEAN LOGIC. NOT Operator Operates on single variable. It gives the complement value of variable.
UNIT4 BOOLEAN LOGIC Boolean algebra is an algebra that deals with Boolean values((true and FALSE). Everyday we have to make logic decisions: Should I carry the book or not?, Should I watch TV or not?
More informationSummary. Boolean Addition
Summary Boolean Addition In Boolean algebra, a variable is a symbol used to represent an action, a condition, or data. A single variable can only have a value of or 0. The complement represents the inverse
More informationReview. EECS Components and Design Techniques for Digital Systems. Lec 05 Boolean Logic 9/404. Seq. Circuit Behavior. Outline.
Review EECS 150  Components and Design Techniques for Digital Systems Lec 05 Boolean Logic 9404 David Culler Electrical Engineering and Computer Sciences University of California, Berkeley Design flow
More informationCircuit analysis summary
Boolean Algebra Circuit analysis summary After finding the circuit inputs and outputs, you can come up with either an expression or a truth table to describe what the circuit does. You can easily convert
More informationR.M.D. ENGINEERING COLLEGE R.S.M. Nagar, Kavaraipettai
L T P C R.M.D. ENGINEERING COLLEGE R.S.M. Nagar, Kavaraipettai 601206 DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING EC8392 UNIT  I 3 0 0 3 OBJECTIVES: To present the Digital fundamentals, Boolean
More informationBawar Abid Abdalla. Assistant Lecturer Software Engineering Department Koya University
Logic Design First Stage Lecture No.5 Boolean Algebra Bawar Abid Abdalla Assistant Lecturer Software Engineering Department Koya University Boolean Operations Laws of Boolean Algebra Rules of Boolean Algebra
More information2008 The McGrawHill Companies, Inc. All rights reserved.
28 The McGrawHill Companies, Inc. All rights reserved. 28 The McGrawHill Companies, Inc. All rights reserved. All or Nothing Gate Boolean Expression: A B = Y Truth Table (ee next slide) or AB = Y 28
More informationBoolean Logic CS.352.F12
Boolean Logic CS.352.F12 Boolean Algebra Boolean Algebra Mathematical system used to manipulate logic equations. Boolean: deals with binary values (True/False, yes/no, on/off, 1/0) Algebra: set of operations
More informationBawar Abid Abdalla. Assistant Lecturer Software Engineering Department Koya University
Logic Design First Stage Lecture No.6 Boolean Algebra Bawar Abid Abdalla Assistant Lecturer Software Engineering Department Koya University Outlines Boolean Operations Laws of Boolean Algebra Rules of
More informationChapter 2: Combinational Systems
Uchechukwu Ofoegbu Chapter 2: Combinational Systems Temple University Adapted from Alan Marcovitz s Introduction to Logic and Computer Design Riddle Four switches can be turned on or off. One is the switch
More informationBOOLEAN ALGEBRA. 1. State & Verify Laws by using :
BOOLEAN ALGEBRA. State & Verify Laws by using :. State and algebraically verify Absorption Laws. (2) Absorption law states that (i) X + XY = X and (ii) X(X + Y) = X (i) X + XY = X LHS = X + XY = X( + Y)
More informationLecture 5. Chapter 2: Sections 47
Lecture 5 Chapter 2: Sections 47 Outline Boolean Functions What are Canonical Forms? Minterms and Maxterms Index Representation of Minterms and Maxterms SumofMinterm (SOM) Representations ProductofMaxterm
More informationIntroduction to Computer Architecture
Boolean Operators The Boolean operators AND and OR are binary infix operators (that is, they take two arguments, and the operator appears between them.) A AND B D OR E We will form Boolean Functions of
More informationClass Subject Code Subject Prepared By Lesson Plan for Time: Lesson. No 1.CONTENT LIST: Introduction to UnitI 2. SKILLS ADDRESSED: Listening I year, 02 sem CS6201 Digital Principles & System Design S.Seedhanadevi
More informationELCT201: DIGITAL LOGIC DESIGN
ELCT201: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 3 Following the slides of Dr. Ahmed H. Madian ذو الحجة 1438 ه Winter
More informationPhiladelphia University Faculty of Information Technology Department of Computer Science. Computer Logic Design. By Dareen Hamoudeh.
Philadelphia University Faculty of Information Technology Department of Computer Science Computer Logic Design By Dareen Hamoudeh Dareen Hamoudeh 1 Canonical Forms (Standard Forms of Expression) Minterms
More informationDIGITAL ELECTRONICS. Vayu Education of India
DIGITAL ELECTRONICS ARUN RANA Assistant Professor Department of Electronics & Communication Engineering Doon Valley Institute of Engineering & Technology Karnal, Haryana (An ISO 9001:2008 ) Vayu Education
More informationENGINEERS ACADEMY. 7. Given Boolean theorem. (a) A B A C B C A B A C. (b) AB AC BC AB BC. (c) AB AC BC A B A C B C.
Digital Electronics Boolean Function QUESTION BANK. The Boolean equation Y = C + C + C can be simplified to (a) (c) A (B + C) (b) AC (d) C. The Boolean equation Y = (A + B) (A + B) can be simplified to
More informationUNIT 2 BOOLEAN ALGEBRA
UNIT 2 BOOLEN LGEBR Spring 2 2 Contents Introduction Basic operations Boolean expressions and truth tables Theorems and laws Basic theorems Commutative, associative, and distributive laws Simplification
More informationGateLevel Minimization
GateLevel Minimization ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2011 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outlines The Map Method
More informationDIGITAL CIRCUIT LOGIC UNIT 9: MULTIPLEXERS, DECODERS, AND PROGRAMMABLE LOGIC DEVICES
DIGITAL CIRCUIT LOGIC UNIT 9: MULTIPLEXERS, DECODERS, AND PROGRAMMABLE LOGIC DEVICES 1 Learning Objectives 1. Explain the function of a multiplexer. Implement a multiplexer using gates. 2. Explain the
More informationLogic and Computer Design Fundamentals. Chapter 2 Combinational Logic Circuits. Part 3 Additional Gates and Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 3 Additional Gates and Circuits Charles Kime & Thomas Kaminski 28 Pearson Education, Inc. (Hyperlinks are active in View
More informationCS February 17
Discrete Mathematics CS 26 February 7 Equal Boolean Functions Two Boolean functions F and G of degree n are equal iff for all (x n,..x n ) B, F (x,..x n ) = G (x,..x n ) Example: F(x,y,z) = x(y+z), G(x,y,z)
More informationGateLevel Minimization
MEC520 디지털공학 GateLevel Minimization JeeHwan Ryu School of Mechanical Engineering GateLevel MinimizationThe Map Method Truth table is unique Many different algebraic expression Boolean expressions may
More informationExperiment 4 Boolean Functions Implementation
Experiment 4 Boolean Functions Implementation Introduction: Generally you will find that the basic logic functions AND, OR, NAND, NOR, and NOT are not sufficient to implement complex digital logic functions.
More informationGateLevel Minimization. section instructor: Ufuk Çelikcan
GateLevel Minimization section instructor: Ufuk Çelikcan Compleity of Digital Circuits Directly related to the compleity of the algebraic epression we use to build the circuit. Truth table may lead to
More information3. The high voltage level of a digital signal in positive logic is : a) 1 b) 0 c) either 1 or 0
1. The number of level in a digital signal is: a) one b) two c) four d) ten 2. A pure sine wave is : a) a digital signal b) analog signal c) can be digital or analog signal d) neither digital nor analog
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Overview Part Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra Standard
More informationSYNERGY INSTITUTE OF ENGINEERING & TECHNOLOGY,DHENKANAL LECTURE NOTES ON DIGITAL ELECTRONICS CIRCUIT(SUBJECT CODE:PCEC4202)
Lecture No:5 Boolean Expressions and Definitions Boolean Algebra Boolean Algebra is used to analyze and simplify the digital (logic) circuits. It uses only the binary numbers i.e. 0 and 1. It is also called
More informationDigital Logic Design. Outline
Digital Logic Design GateLevel Minimization CSE32 Fall 2 Outline The Map Method 2,3,4 variable maps 5 and 6 variable maps (very briefly) Product of sums simplification Don t Care conditions NAND and NOR
More information數位系統 Digital Systems 朝陽科技大學資工系. Speaker: FuwYi Yang 楊伏夷. 伏夷非征番, 道德經察政章 (Chapter 58) 伏者潛藏也道紀章 (Chapter 14) 道無形象, 視之不可見者曰夷
數位系統 Digital Systems Department of Computer Science and Information Engineering, Chaoyang University of Technology 朝陽科技大學資工系 Speaker: FuwYi Yang 楊伏夷 伏夷非征番, 道德經察政章 (Chapter 58) 伏者潛藏也道紀章 (Chapter 14) 道無形象,
More informationUNIT II. Circuit minimization
UNIT II Circuit minimization The complexity of the digital logic gates that implement a Boolean function is directly related to the complexity of the algebraic expression from which the function is implemented.
More informationCode No: 07A3EC03 Set No. 1
Code No: 07A3EC03 Set No. 1 II B.Tech I Semester Regular Examinations, November 2008 SWITCHING THEORY AND LOGIC DESIGN ( Common to Electrical & Electronic Engineering, Electronics & Instrumentation Engineering,
More informationEE292: Fundamentals of ECE
EE292: Fundamentals of ECE Fall 2012 TTh 10:0011:15 SEB 1242 Lecture 22 121115 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Review Binary Number Representation Binary Arithmetic Combinatorial Logic
More informationCombinational Logic. Prof. Wangrok Oh. Dept. of Information Communications Eng. Chungnam National University. Prof. Wangrok Oh(CNU) 1 / 93
Combinational Logic Prof. Wangrok Oh Dept. of Information Communications Eng. Chungnam National University Prof. Wangrok Oh(CNU) / 93 Overview Introduction 2 Combinational Circuits 3 Analysis Procedure
More informationContents. Chapter 3 Combinational Circuits Page 1 of 34
Chapter 3 Combinational Circuits Page of 34 Contents Contents... 3 Combinational Circuits... 2 3. Analysis of Combinational Circuits... 2 3.. Using a Truth Table... 2 3..2 Using a Boolean unction... 4
More informationPresented By : Alok Kumar Lecturer in ECE C.R.Polytechnic, Rohtak
Presented By : Alok Kumar Lecturer in ECE C.R.Polytechnic, Rohtak Content  Introduction 2 Feature 3 Feature of BJT 4 TTL 5 MOS 6 CMOS 7 K Map  Introduction Logic IC ASIC: Application Specific
More informationCode No: R Set No. 1
Code No: R059210504 Set No. 1 II B.Tech I Semester Regular Examinations, November 2006 DIGITAL LOGIC DESIGN ( Common to Computer Science & Engineering, Information Technology and Computer Science & Systems
More informationComputer Science. Unit4: Introduction to Boolean Algebra
Unit4: Introduction to Boolean Algebra Learning Objective At the end of the chapter students will: Learn Fundamental concepts and basic laws of Boolean algebra. Learn about Boolean expression and will
More informationB.Tech II Year I Semester (R13) Regular Examinations December 2014 DIGITAL LOGIC DESIGN
B.Tech II Year I Semester () Regular Examinations December 2014 (Common to IT and CSE) (a) If 1010 2 + 10 2 = X 10, then X is  Write the first 9 decimal digits in base 3. (c) What is meant by don
More informationQUESTION BANK FOR TEST
CSCI 2121 Computer Organization and Assembly Language PRACTICE QUESTION BANK FOR TEST 1 Note: This represents a sample set. Please study all the topics from the lecture notes. Question 1. Multiple Choice
More informationDigital Logic Design Exercises. Assignment 1
Assignment 1 For Exercises 15, match the following numbers with their definition A Number Natural number C Integer number D Negative number E Rational number 1 A unit of an abstract mathematical system
More informationThis presentation will..
Component Identification: Digital Introduction to Logic Gates and Integrated Circuits Digital Electronics 2014 This presentation will.. Introduce transistors, logic gates, integrated circuits (ICs), and
More informationComputer Organization
Computer Organization (Logic circuits design and minimization) KR Chowdhary Professor & Head Email: kr.chowdhary@gmail.com webpage: krchowdhary.com Department of Computer Science and Engineering MBM Engineering
More informationBoolean Algebra & Digital Logic
Boolean Algebra & Digital Logic Boolean algebra was developed by the Englishman George Boole, who published the basic principles in the 1854 treatise An Investigation of the Laws of Thought on Which to
More informationTWOLEVEL COMBINATIONAL LOGIC
TWOLEVEL COMBINATIONAL LOGIC OVERVIEW Canonical forms Tolevel simplification Boolean cubes Karnaugh maps QuineMcClusky (Tabulation) Method Don't care terms Canonical and Standard Forms Minterms and
More informationChap.3 3. Chap reduces the complexity required to represent the schematic diagram of a circuit Library
3.1 Combinational Circuits 2 Chap 3. logic circuits for digital systems: combinational vs sequential Combinational Logic Design Combinational Circuit (Chap 3) outputs are determined by the present applied
More informationVALLIAMMAI ENGINEERING COLLEGE. SRM Nagar, Kattankulathur DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING EC6302 DIGITAL ELECTRONICS
VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur603 203 DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING EC6302 DIGITAL ELECTRONICS YEAR / SEMESTER: II / III ACADEMIC YEAR: 20152016 (ODD
More informationCode No: R Set No. 1
Code No: R059210504 Set No. 1 II B.Tech I Semester Supplementary Examinations, February 2007 DIGITAL LOGIC DESIGN ( Common to Computer Science & Engineering, Information Technology and Computer Science
More informationDigital Logic Design (CEN120) (3+1)
Digital Logic Design (CEN120) (3+1) ASSISTANT PROFESSOR Engr. Syed Rizwan Ali, MS(CAAD)UK, PDG(CS)UK, PGD(PM)IR, BS(CE)PK HEC Certified Master Trainer (MTFPDP) PEC Certified Professional Engineer (COM/2531)
More informationCombinational Logic II
Combinational Logic II Ranga Rodrigo July 26, 2009 1 Binary AdderSubtractor Digital computers perform variety of information processing tasks. Among the functions encountered are the various arithmetic
More informationHenry Lin, Department of Electrical and Computer Engineering, California State University, Bakersfield Lecture 7 (Digital Logic) July 24 th, 2012
Henry Lin, Department of Electrical and Computer Engineering, California State University, Bakersfield Lecture 7 (Digital Logic) July 24 th, 2012 1 Digital vs Analog Digital signals are binary; analog
More informationNumber Systems UNIT. Learning Objectives. 1.0 Introduction
UNIT 1 Number Systems Learning Objectives To study Binary, Octal, Hexadecimal, Decimal number systems. Conversion of Binary to Octal, Binary to decimal, Binary to Hexa decimal and Conversion. Binary Addition,
More informationMGUBCA205 Second Sem Core VI Fundamentals of Digital Systems MCQ s. 2. Why the decimal number system is also called as positional number system?
MGUBCA205 Second Sem Core VI Fundamentals of Digital Systems MCQ s Unit1 Number Systems 1. What does a decimal number represents? A. Quality B. Quantity C. Position D. None of the above 2. Why the
More informationMidterm Exam Review. CS 2420 :: Fall 2016 Molly O'Neil
Midterm Exam Review CS 2420 :: Fall 2016 Molly O'Neil Midterm Exam Thursday, October 20 In class, pencil & paper exam Closed book, closed notes, no cell phones or calculators, clean desk 20% of your final
More informationStandard Forms of Expression. Minterms and Maxterms
Standard Forms of Expression Minterms and Maxterms Standard forms of expressions We can write expressions in many ways, but some ways are more useful than others A sum of products (SOP) expression contains:
More informationCHAPTER2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, KMap and QuineMcCluskey
CHAPTER2 STRUCTURE OF BOOLEAN FUNCTION USING GATES, KMap and QuineMcCluskey 2. Introduction Logic gates are connected together to produce a specified output for certain specified combinations of input
More informationDesigning Computer Systems Boolean Algebra
Designing Computer Systems Boolean Algebra 08:34:45 PM 4 June 2013 BA1 Scott & Linda Wills Designing Computer Systems Boolean Algebra Programmable computers can exhibit amazing complexity and generality.
More informationChapter 2 Combinational
Computer Engineering 1 (ECE290) Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization HOANG Trang 2008 Pearson Education, Inc. Overview Part 1 Gate Circuits and Boolean Equations Binary Logic
More information01 Introduction to Digital Logic. ENGR 3410 Computer Architecture Mark L. Chang Fall 2006
Introduction to Digital Logic ENGR 34 Computer Architecture Mark L. Chang Fall 26 Acknowledgements Patterson & Hennessy: Book & Lecture Notes Patterson s 997 course notes (U.C. Berkeley CS 52, 997) Tom
More informationCS6201 DIGITAL PRINCIPLES AND SYSTEM DESIGN Lecture Notes
CS6201 DIGITAL PRINCIPLES AND SYSTEM DESIGN Lecture Notes 1.1 Introduction: UNIT I BOOLEAN ALGEBRA AND LOGIC GATES Like normal algebra, Boolean algebra uses alphabetical letters to denote variables. Unlike
More information