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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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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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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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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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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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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

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Logic Diagrams and Expressions

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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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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 2

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Proof

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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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Logic Diagram

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Useful Theorems

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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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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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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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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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Minterms

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Maxterms

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Minterms & Maxterms for 2 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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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-Minterm Examples

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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