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Binary Variables, слайд №1Binary Variables, слайд №2Binary Variables, слайд №3Binary Variables, слайд №4Binary Variables, слайд №5Binary Variables, слайд №6Binary Variables, слайд №7Binary Variables, слайд №8Binary Variables, слайд №9Binary Variables, слайд №10Binary Variables, слайд №11Binary Variables, слайд №12Binary Variables, слайд №13Binary Variables, слайд №14Binary Variables, слайд №15Binary Variables, слайд №16Binary Variables, слайд №17Binary Variables, слайд №18Binary Variables, слайд №19Binary Variables, слайд №20Binary Variables, слайд №21Binary Variables, слайд №22Binary Variables, слайд №23Binary Variables, слайд №24Binary Variables, слайд №25Binary Variables, слайд №26Binary Variables, слайд №27Binary Variables, слайд №28Binary Variables, слайд №29Binary Variables, слайд №30Binary Variables, слайд №31Binary Variables, слайд №32Binary Variables, слайд №33Binary Variables, слайд №34Binary Variables, слайд №35Binary Variables, слайд №36Binary Variables, слайд №37Binary Variables, слайд №38Binary Variables, слайд №39Binary Variables, слайд №40Binary Variables, слайд №41Binary Variables, слайд №42Binary Variables, слайд №43Binary Variables, слайд №44Binary Variables, слайд №45Binary Variables, слайд №46Binary Variables, слайд №47Binary Variables, слайд №48Binary Variables, слайд №49Binary Variables, слайд №50Binary Variables, слайд №51Binary Variables, слайд №52Binary Variables, слайд №53Binary Variables, слайд №54Binary Variables, слайд №55Binary Variables, слайд №56
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Binary Variables Recall that the two binary values have different names: True/False On/Off Yes/No 1/0 We use 1 and 0 to denote the two values.
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Binary Variables Recall that the two binary values have different names: True/False On/Off Yes/No 1/0 We use 1 and 0 to denote the two values.

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Boolean Algebra Invented by George Boole in 1854 An algebraic structure defined by a set B = {0, 1}, together with two binary operators (+ and ·) and a unary operator ( ¯ ),
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Boolean Algebra Invented by George Boole in 1854 An algebraic structure defined by a set B = {0, 1}, together with two binary operators (+ and ·) and a unary operator ( ¯ ),

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Binary Logic and Gates Binary variables take on one of two values. Logical operators operate on binary values and binary variables. Basic logical operators are the logic functions AND, OR and NOT. Logic gates implement logic functions. Boolean Algebra: a useful mathematical system for specifying and transforming logic functions. We study Boolean algebra as a foundation for designing and analyzing digital systems!
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Binary Logic and Gates Binary variables take on one of two values. Logical operators operate on binary values and binary variables. Basic logical operators are the logic functions AND, OR and NOT. Logic gates implement logic functions. Boolean Algebra: a useful mathematical system for specifying and transforming logic functions. We study Boolean algebra as a foundation for designing and analyzing digital systems!

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Logic Gates In the earliest computers, switches were opened and closed by magnetic fields produced by energizing coils in relays. The switches in turn opened and closed the current paths. Later, vacuum tubes that open and close current paths electronically replaced relays. Today, transistors are used as electronic switches that open and close current paths.
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Logic Gates In the earliest computers, switches were opened and closed by magnetic fields produced by energizing coils in relays. The switches in turn opened and closed the current paths. Later, vacuum tubes that open and close current paths electronically replaced relays. Today, transistors are used as electronic switches that open and close current paths.

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Logical Operations The three basic logical operations are: AND OR NOT AND is denoted by a dot (·). OR is denoted by a plus (+). NOT is denoted by an overbar ( ¯ ), a single quote mark (') after, or (~) before the variable.
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Logical Operations The three basic logical operations are: AND OR NOT AND is denoted by a dot (·). OR is denoted by a plus (+). NOT is denoted by an overbar ( ¯ ), a single quote mark (') after, or (~) before the variable.

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Truth Tables
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Truth Tables

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Operator Definitions
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Operator Definitions

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Produce a truth table l In the BooleanAlgebra, verify using truth table that (X + Y)’ = X’Y’ In the Boolean Algebra, verify using truth table that X + XY = X
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Produce a truth table l In the BooleanAlgebra, verify using truth table that (X + Y)’ = X’Y’ In the Boolean Algebra, verify using truth table that X + XY = X

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1. Write the boolean expression for the below circuit
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1. Write the boolean expression for the below circuit

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2. Write the boolean expression for the below circuit
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2. Write the boolean expression for the below circuit

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Problem 1 A system used 3 switches A,B and C; a combination of switches determines whether an alarm, X, sounds: If switch A or Switch B are in the ON position and if switch C is in the OFF position then a signal to sound an alarm, X is produced. Convert this problem into a logic statement.
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Problem 1 A system used 3 switches A,B and C; a combination of switches determines whether an alarm, X, sounds: If switch A or Switch B are in the ON position and if switch C is in the OFF position then a signal to sound an alarm, X is produced. Convert this problem into a logic statement.

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Problem 2 A nuclear power station has a safety system based on three inputs to a logic circuit(network). A warning signal ( S = 1) is produced when certain conditions in the nuclear power station occur based on these three inputs A warning signal (S=1) will be produced when any of the following occurs. Either (a) Temperature > 115 C and Cooling water <=120 litres/hour or (b) Temperature <=115 C and when Reactor pressure > 15 bar or cooling water <= 120 litres/hour Draw a logic circuit and truth table to show all the possible situations when the warning signal (S) could be received.
Описание слайда:
Problem 2 A nuclear power station has a safety system based on three inputs to a logic circuit(network). A warning signal ( S = 1) is produced when certain conditions in the nuclear power station occur based on these three inputs A warning signal (S=1) will be produced when any of the following occurs. Either (a) Temperature > 115 C and Cooling water <=120 litres/hour or (b) Temperature <=115 C and when Reactor pressure > 15 bar or cooling water <= 120 litres/hour Draw a logic circuit and truth table to show all the possible situations when the warning signal (S) could be received.

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Logic Diagrams and Expressions
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Logic Diagrams and Expressions

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Boolean Algebra
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Boolean Algebra

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Some Properties of Boolean Algebra Boolean Algebra is defined in general by a set B that can have more than two values A two-valued Boolean algebra is also know as Switching Algebra. The Boolean set B is restricted to 0 and 1. Switching circuits can be represented by this algebra. The dual of an algebraic expression is obtained by interchanging + and · and interchanging 0’s and 1’s. The identities appear in dual pairs. When there is only one identity on a line the identity is self-dual, i. e., the dual expression = the original expression. Sometimes, the dot symbol ‘’ (AND operator) is not written when the meaning is clear.
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Some Properties of Boolean Algebra Boolean Algebra is defined in general by a set B that can have more than two values A two-valued Boolean algebra is also know as Switching Algebra. The Boolean set B is restricted to 0 and 1. Switching circuits can be represented by this algebra. The dual of an algebraic expression is obtained by interchanging + and · and interchanging 0’s and 1’s. The identities appear in dual pairs. When there is only one identity on a line the identity is self-dual, i. e., the dual expression = the original expression. Sometimes, the dot symbol ‘’ (AND operator) is not written when the meaning is clear.

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Dual of a Boolean Expression Example: F = (A + C) · B + 0 dual F = (A · C + B) · 1 = A · C + B Example: G = X · Y + (W + Z) dual G = (X+Y) · (W · Z) = (X+Y) · (W+Z) Example: H = A · B + A · C + B · C dual H = (A+B) · (A+C) · (B+C)
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Dual of a Boolean Expression Example: F = (A + C) · B + 0 dual F = (A · C + B) · 1 = A · C + B Example: G = X · Y + (W + Z) dual G = (X+Y) · (W · Z) = (X+Y) · (W+Z) Example: H = A · B + A · C + B · C dual H = (A+B) · (A+C) · (B+C)

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Boolean Algebraic Proof – Example 1 A + A · B = A (Absorption Theorem) Proof Steps Justification A + A · B = A · 1 + A · B Identity element: A · 1 = A = A · ( 1 + B) Distributive = A · 1 1 + B = 1 = A Identity element Our primary reason for doing proofs is to learn: Careful and efficient use of the identities and theorems of Boolean algebra, and How to choose the appropriate identity or theorem to apply to make forward progress, irrespective of the application.
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Boolean Algebraic Proof – Example 1 A + A · B = A (Absorption Theorem) Proof Steps Justification A + A · B = A · 1 + A · B Identity element: A · 1 = A = A · ( 1 + B) Distributive = A · 1 1 + B = 1 = A Identity element Our primary reason for doing proofs is to learn: Careful and efficient use of the identities and theorems of Boolean algebra, and How to choose the appropriate identity or theorem to apply to make forward progress, irrespective of the application.

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Boolean Algebraic Proof – Example 2
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Boolean Algebraic Proof – Example 2

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Proof
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Proof

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Minimization of Boolean Expression
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Minimization of Boolean Expression

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Simplification of Boolean Algebra (A + B)(A + C) = A + BC This rule can be proved as follows: (A + B)(A + C) = AA + AC + AB + BC( Distributive law) = A + AC + AB + BC ( AA = A) = A( 1 + C) + AB + BC (1 + C = 1) = A. 1 + AB + BC = A(1 + B) + BC (1 + B = 1) = A. 1 + BC ( A . 1 = A) = A + BC
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Simplification of Boolean Algebra (A + B)(A + C) = A + BC This rule can be proved as follows: (A + B)(A + C) = AA + AC + AB + BC( Distributive law) = A + AC + AB + BC ( AA = A) = A( 1 + C) + AB + BC (1 + C = 1) = A. 1 + AB + BC = A(1 + B) + BC (1 + B = 1) = A. 1 + BC ( A . 1 = A) = A + BC

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Logic Diagram
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Logic Diagram

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Useful Theorems
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Useful Theorems

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De morgan’s Law
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De morgan’s Law

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Gate equivalencies and the corresponding truth tables that illustrate De Morgan's theorems.
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Gate equivalencies and the corresponding truth tables that illustrate De Morgan's theorems.

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Truth Table to Verify De Morgan’s
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Truth Table to Verify De Morgan’s

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Simplification-Example Using Boolean algebra techniques, simplify this expression: AB + A(B + C) + B(B + C) Step 1: Apply the distributive law to the second and third terms in the expression, as follows: AB + AB + AC + BB + BC Step 2: Apply (BB = B) to the fourth term. AB + AB + AC + B + BC Step 3: Apply (AB + AB = AB) to the first two terms. AB + AC + B + BC Step 4: Apply (B + BC = B) to the last two terms. AB + AC + B Step 5: Apply (AB + B = B) to the first and third terms. B+AC
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Simplification-Example Using Boolean algebra techniques, simplify this expression: AB + A(B + C) + B(B + C) Step 1: Apply the distributive law to the second and third terms in the expression, as follows: AB + AB + AC + BB + BC Step 2: Apply (BB = B) to the fourth term. AB + AB + AC + B + BC Step 3: Apply (AB + AB = AB) to the first two terms. AB + AC + B + BC Step 4: Apply (B + BC = B) to the last two terms. AB + AC + B Step 5: Apply (AB + B = B) to the first and third terms. B+AC

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Binary Variables, слайд №28
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Truth Tables – Cont’d
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Truth Tables – Cont’d

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Logic Diagram
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Logic Diagram

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Logic Diagram
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Logic Diagram

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Logic Diagram
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Logic Diagram

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Logic Diagram
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Logic Diagram

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Expression Simplification
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Expression Simplification

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Canonical Forms….. Minterms and Maxterms Sum-of-products (SOP) Canonical Form Product-of-sum (POS) Canonical Form Representation of Complements of Functions Conversions between Representations
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Canonical Forms….. Minterms and Maxterms Sum-of-products (SOP) Canonical Form Product-of-sum (POS) Canonical Form Representation of Complements of Functions Conversions between Representations

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Minterms
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Minterms

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Maxterms
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Maxterms

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Minterms & Maxterms for 2 variables
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Minterms & Maxterms for 2 variables

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Minterms & Maxterms for 3 variables
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Minterms & Maxterms for 3 variables

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The Standard SOP Form A standard SOP expression is one in which all the variables in the domain appear in each product term in the expression. Example: Standard SOP expressions are important in: Constructing truth tables The Karnaugh map simplification method
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The Standard SOP Form A standard SOP expression is one in which all the variables in the domain appear in each product term in the expression. Example: Standard SOP expressions are important in: Constructing truth tables The Karnaugh map simplification method

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Converting Product Terms to Standard SOP (example) Convert the following Boolean expression into standard SOP form:
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Converting Product Terms to Standard SOP (example) Convert the following Boolean expression into standard SOP form:

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Sum-Of- Product (SOP)
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Sum-Of- Product (SOP)

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Sum-Of-Minterm Examples
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Sum-Of-Minterm Examples

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Implementation of an SOP AND/OR implementation
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Implementation of an SOP AND/OR implementation

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The Standard POS Form A standard POS expression is one in which all the variables in the domain appear in each sum term in the expression. Example: Standard POS expressions are important in: Constructing truth tables The Karnaugh map simplification method
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The Standard POS Form A standard POS expression is one in which all the variables in the domain appear in each sum term in the expression. Example: Standard POS expressions are important in: Constructing truth tables The Karnaugh map simplification method

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Converting a Sum Term to Standard POS (example) Convert the following Boolean expression into standard POS form:
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Converting a Sum Term to Standard POS (example) Convert the following Boolean expression into standard POS form:

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Product-Of-Maxterm (POM)
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Product-Of-Maxterm (POM)

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Product-Of-Maxterm Examples
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Product-Of-Maxterm Examples

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Converting to Sum-of-Minterms Form
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Converting to Sum-of-Minterms Form

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Converting to Product-of-Maxterms Form
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Converting to Product-of-Maxterms Form

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Conversions Between Canonical Forms
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Conversions Between Canonical Forms

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Standard Sum-of-Products (SOP)
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Standard Sum-of-Products (SOP)

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Three-way light control
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Three-way light control

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Three-way light control
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Three-way light control

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Car safety alarm
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Car safety alarm

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Binary Variables, слайд №56
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