UploadLanguage (EN)Support
BusinessMobileSocial MediaMarketingTechnologyArt & PhotosCareerDesignEducationPresentations & Public SpeakingGovernment & NonprofitHealthcareInternetLawLeadership & ManagementAutomotiveEngineeringSoftwareRecruiting & HRRetailSalesServicesScienceSmall Business & EntrepreneurshipFoodEnvironmentEconomy & FinanceData & AnalyticsInvestor RelationsSportsSpiritualNews & PoliticsTravelSelf ImprovementReal EstateEntertainment & HumorHealth & MedicineDevices & HardwareLifestyle
Upload
Sign in
Uploaded byTamiratDejene1
PDF, PPTX170 views

DLD Chapter-2.pdf

The document discusses Boolean algebra and logic gates. It begins with an introduction to Boolean algebra and how it provides a mathematical framework for digital electronic systems. It then covers topics such as Boolean variables, expressions, logic gates like AND, OR, NOT, NAND, NOR, XOR and XNOR. It defines Boolean functions and discusses ways of representing functions using truth tables, logic diagrams and algebraic expressions. It also covers concepts like positive and negative logic, postulates and theorems of Boolean algebra, Venn diagrams, canonical and standard forms of logic functions including sum of products and product of sums forms.

Education◦
Download as PDF, PPTX
1 / 58
2 / 58
3 / 58
4 / 58
5 / 58
6 / 58
7 / 58
8 / 58
9 / 58
10 / 58
11 / 58
12 / 58
13 / 58
14 / 58
15 / 58
16 / 58
17 / 58
18 / 58
19 / 58
20 / 58
Most read
2.1 Under graduate (2016) Girma Adam (M.Tech) Chapter Two Boolean Algebra and Logic gates
2.2 Topics discussed in this section: Cont’d.. Introduction Digital Logic Gates Logic Operators Positive and Negative Logic Postulates and Theorems of Boolean Algebra Venn Diagram Boolean Functions Simplification of Boolean Expressions Canonical and Standard forms of logic functions
2.3 Introduction  Digital electronic systems manipulate binary information.  To design such systems we need a convenient mathematical framework, this framework provides Boolean algebra  Boolean algebra:- is a mathematical system for the manipulation of variables that can have one of two values. - In formal logic, these values are “true” and “false.” - In digital systems, these values are “on” and “off,” 1 and 0, or “high” and “low.”  Boolean Variable :- the input and output variables of a digital systems.
2.4 Cont’d.. - We use symbols to represent Boolean variables (e.g.: A, B, C, X, Y, Z) - A Boolean Variable takes the value of either 0 or 1 - Boolean 0 and 1 correspond to binary 0 and 1.  Boolean expressions:- are created by performing operations on Boolean variables. - Common Boolean operators include AND, OR, and NOT.
2.5 Basic Logic Gates & Boolean Operators  Gates got their name from their function of allowing or blocking (gating) the flow of digital information.  A gate has one or more inputs and produces an output depending on the input(s).  There are elementary logic gates, each gate has its own logic symbol which allows complex functions to be represented by a logic diagram.  A Boolean operator of a logic gate can be completely described using a truth table.
2.6 Cont’d..  The function of each gate can be represented by a truth table or using Boolean notation. • Inverter NOR gate • AND gate Exclusive-OR gate • OR gate Exclusive-NOR gate . NAND gate logic buffer gate
2.7 Cont,d..  The NOT gate (or inverter)  The AND gate
2.8 Cont,d..  The OR gate  The NAND gate
2.9 Cont’d..  The NOR gate  The Exclusive OR gate
2.10 Cont’d..  The Exclusive NOR gate  A logic buffer gate
2.11 Cont,d..  The operator symbols ∧ ,∨,￢ are not available on standard keyboards  To make things easier to type, in digital electronics we use 
2.12 Example  To Show how basic logic gates can be constructed. We will also examine ways to simplify complicated circuits. 1. Consider the following lighting circuit with two switches S1 and S2. Under what conditions will the bulb light ?
2.13 Cont’d..  This problem is to realise that the bulb will light ONLY when both switches A AND B are closed. This can be represented in a Truth Table. Switch A Switch B Output Y Open Open NO Open Closed NO Closed Open NO Closed Closed YES  This type of truth table is called an AND table. A B Y 0 0 0 0 1 0 1 0 0 1 1 1
2.14 Cont’d.. 2. Two switches in parallel
2.15 Cont’d.. 3.Three switches in series
2.16 Cont’d.. 4. Three switches in parallel
2.17 Cont’d.. 5. A series/parallel arrangement
2.18 Cont’d.. 6. OR gate in alarm system
2.19 Cont’d.. 8. 4 input OR gate
2.20 Cont’d.. Summary of OR operation 1. Output value will be:-  1 whenever any input is 1  0 otherwise 2. An OR gate is a logic circuit that performs an OR (sum) operation on the circuit’s inputs 3. The Boolean expression x = A + B is read as “x equals A OR B”
2.21 Cont’d.. 9. 3 input And gate operation
2.22 Cont’d.. Summary of AND operation 1. Output value will be:-  1 whenever all input is 1  0 otherwise 2. An AND gate is a logic circuit that performs an and (product) operation on the circuit’s inputs 3. The Boolean expression x = A . B is read as “x equals A and B”
2.23 Cont’d..
2.24 Positive and Negative Logic  The binary variables, as we know, can have either of the two states, i.e. 1. the logic ‘0’ state or 2. the logic ‘1’ state.  These logic states in digital systems are represented by two different voltage levels or two different current levels.  If the more positive of the two voltage or current levels represents a logic ‘1’ and the less positive of the two levels represents a logic ‘0’, then the logic system is referred to as a positive logic system.
2.25 Cont’d..  If the more positive of the two voltage or current levels represents a logic ‘0’ and the less positive of the two levels represents a logic ‘1’, then the logic system is referred to as a negative logic system. In positive logic system In negative logic system.
2.26 Cont’d..  That is, OR gate hardware in the positive logic system behaves like an AND gate in the negative logic system. The reverse is also true. Similarly, a positive NOR is a negative NAND, and vice versa.
2.27 Postulates and Theorems of Boolean Algebra  Postulates form the basic assumptions from which it is possible to deduce the rules, theorems, and properties of the system. Postulate 1: Closure - A Boolean algebra is a closed algebraic system containing a set of two or more elements and two operators +(OR Operator) and . (AND Operator). Postulate 2: Identity Property - There exists unique elements 1 (one) and 0 (zero) such that: (a) x+0=x (b) x.1=x
2.28 Cont’d.. - where 0 is the identity element for the OR operator and 1 is the identity element for the AND operator. Postulate 3: Commutative - For every x and y in set (a) x+y=y+x (b) x.y=y.x Postulate 4: Associative - For every x , y and Z in set (a) x+(y+z)=(x+y)+z (b) x.(y.z) =(x.y).z
2.29 Cont’d.. Postulate 5: Distributive Property - For every x, y and Z in set (a) x.(y+z)=x.y+x.z (b) x+y.z =(x+y).(x+z) Postulate 6: Complement Property - For every x in set there exists a unique element x such that: (a) X + X = 1 (b) X . X = 0
2.30 Basic Boolean Theorems Theorem 1: (a) a + a = a (b) a . a = a Proof a + a = (a + a)1 P2b aa = aa + 0 p2a = (a + a)(a + a) p6a = (aa) + (aa) p6b = a +aa p5b = (a)(a + a) p4a = a + 0 p6b = a . 1 p6a = a p2a = a p2b
2.31 Cont’d..  Theorem 2: For each X in B: (a) X + 1 = 1 (b) X . 0 = 0  Proof of (a): X + 1 = 1 . (X + 1) = (X + X’) . (X + 1) = X + (X’. 1) = X + X’ = 1  Proof of (b): X . 0 = 0 + (X . 0) = (X . X’) + (X . 0) = X . (X’ + 0) = X . X’ = 0
2.32 Cont’d..  Theorem 3. X’’= X Proof X’’ = X’’ + 0 = X’’ + X . X’ = (X’’ + X) . (X’’ + X’) = (X’’ + X) . 1 = (X’’ + X) . (X’ + X) = X + (X’ . X’’) = X + 0 = X
2.33 Cont’d.. Theorem 5: Absorption a + ab = a + b a(a + b) = ab Theorem 6: ab + ab = a (a + b)(a + b) = a Theorem 7: ab + abc = ab + ac (a + b)(a + b + c) = (a + b)(a + c)
2.34 Cont’d... Theorem 8: DeMorgan’s theorem a + b = ab ab = a + b Theorem 9: Consensus ab + ac + bc = ab + ac (a + b) + (a + c) + (b + c) = (a + b) + (a + c)
2.35 Cont’d.. Example 1: simplify the Boolean expression F = ab + ab + ab Solution
2.36 Summary
2.37 Summary
2.38 Venn Diagram  Venn diagram:-is a diagram which helpful illustration that may be used to visualize the relationships among the variables of a Boolean expression
2.39 Boolean function  A Boolean function :- is a Boolean expression formed with binary variables ,operators , parenthesis and equal sign  A Boolean function has: • At least one Boolean variable, • At least one Boolean operator, and • At least one input from the set {0,1}.  Define f (a1, a2,a3,.........,an) Where f = Boolean function with value of 0 or 1 a1, a2, a3, …. = Boolean variables (0 or 1)
2.40 Cont’d..  Logical functions can be represented in two ways: • A finite, but non-unique Boolean expression • A unique and finite truth tables  Example of Boolean function or expression: f(x,y,z) = (x + y’)z + x’  Notations: • f is the name of the function. • (x,y,z) are the input variables, each representing 1 or 0. • A literal is any occurrence of an input variable or its complement. The function above has four literals: x, y’, z, x’. 4 literals
2.41 Cont’d..  Precedence is important 1. First, perform all inversions of single terms 2. Perform all operations within parentheses 3. Do AND before OR unless parentheses indicate . 4. If an expression has a bar over it (inversion), calculate the expression first, then invert the result. 5. NOT has the highest precedence, followed by AND, and then OR.
2.42 Cont’d..  To represent a function in a truth table, With n variables there are 2n input combinations.  A Boolean function may be transformed from an algebraic expression in to a logical diagram through composition of logic gates.
2.43 Implementation of Boolean function with Gates Example 1:- Implement the function Example2: Implement the function
2.44 Cont’d.. Example 1:- Find the truth table OR A Y NOT AND B C AND 3# of Inputs = # of Combinations 2 3 = 8
2.45 Generating a Boolean expression from logic diagrams Example 1:- Generate the Boolean expression from the logic gate Example 2:
2.46 Cont,d.. Example 3:- Generate the Boolean expression from the logic gate and find the truth table
2.47 Canonical and Standard forms of logic functions 1. Minterm and Maxterm  A binary number may appear either in its normal form (x) or in its complement form (x’).  Now consider two binary variables x and y combined with an AND operation, since each variable may appear in either form, there are four possible combinations (x’y’, x’y, xy’ and xy).  Each of these four AND terms is called a minterm or standard product.
2.48 Cont’d..  Each minterm is obtained from an AND term of the n variables, a symbol for each minterm is the form mj, where j denotes the decimal equivalent of the binary number of the minterm designated  In a similar fashion, n variables forming an OR term, with each variables being , or complement or un complement, provides 2n possible combinations, called maxiterms or standard sum  Note that each maxterm is the complement of its corresponding minterm, and vice versa.  Boolean functions expressed as a sum of minterms or products of maxterms are said to be in Canonical form
2.49 Cont’d..
2.50 Cont’d..  There are two canonical forms for Boolean expressions: 1. Sum-of-products (SOP):-are Boolean functions that formed by SUMMING ANDed terms. 2. product-of-sums (POS):- are Boolean functions that formed by taking the PRODUCT of Ored terms.
2.51 Canonical SOP Form 1. Minterm: • A product term which contains exactly one complemented or un complemented form of each variable 2. Canonical SOP form • A function which is a sum of only minterms Decimal No a b c Minterms Notation 0 0 0 0 a b c m0 1 0 0 1 a b c m1 2 0 1 0 a b c m2 3 0 1 1 a b c m3 4 1 0 0 a b c m4 5 1 0 1 a b c m5 6 1 1 0 a b c m6 7 1 1 1 a b c m7
2.52 Cont’d..  Minterms are equal to 1 when true.  mj is the decimal equivalent of binary code (a,b,c)  A Boolean function may be expressed algebraically from a given truth table by forming a minterm for each combination of the variables that produces a one in the function, and then taking the OR of all those terms. Example:- F ( A, B,C ) = ABC + ABC + ABC This can be expressed as: m1+m4+m5 F ( A, B,C) =Σm(1,4,5)
2.53 Cont’d.. A B C F 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 1 1 0 m1 = m4 = m5 = Find F F =Σm(0,2,6,7), all 0’s in the truth table
2.54 Canonical POS Form 1.Maxterms • A sum term which contains exactly one complemented or un complemented form of each of input variable. 2. Canonical POS form • A function which is a product of only maxterms Maxterms are equal to 0 when true Decimal No a b c Minterms Notation 0 0 0 0 a+ b+ c M0 1 0 0 1 a+ b+ c M1 2 0 1 0 a +b +c M2 3 0 1 1 a +b+ c M3 4 1 0 0 a +b+ c M4 5 1 0 1 a +b +c M5 6 1 1 0 a +b+ c M6 7 1 1 1 a +b +c M7
2.55 Example F = (A+ B +C)(A+ B +C)(A+ B +C) Or shortly a:- F = M1.M6.M2 , F ( A, B,C) =Π M (1,2,6) A B C F 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 1 1 M1 = A+ B +C M2 = A+ B +C M2 = A+ B +C
2.56 Cont’d.. Find F F = Π M (0,3,4,5,7), 1’s in the truth table.  General: Maxterms are complements of minterms. Converting SOP function from non-canonical to canonical form 1. Express the Boolean function F A+B’C in a sum minterms. Solution - The function has three variables, A, B and C. The first term A is Missing two variables; therefore
2.57 Cont’d.. - This is still missing one variable - The second term B’C is missing one variable - Combine all terms, we have
2.58 Cont’d..