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Ajman

PPT ON BOOLEAN ALGEBRA

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Published in: Physics
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It’s a convenient way and systematic way of expressing and analyzing the operation of logic circuits.

Aswathi K / Abu Dhabi

4 years of teaching experience

Qualification: Bachelor of Engineering in AE&I

Teaches: Electronics, Physics, Engineering, Electrical Technology, Biology, Social Studies, Maths, Chemistry, Mathematics

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  1. Boolean Algebra
  2. Introduction 1854: Loeical aleebra was published by George Boole -5 known today as "Boolean Algebra" It's a convenient way and systematic way of expressing and analyzing the operation of logic circuits. 1938: Claude Shannon was the first to apply Boole's work to the analysis and design of logic circuits.
  3. Boolean Operations & Expressions Variable — a symbol used to represent a logical quantity. Complement — the inverse of a variable and is indicated by a bar over the variable. Literal— a variable or the complement of a variable.
  4. Boolean Addition Boolean addition is equivalent to the OR operation A A sum term is produced by an OR operation with no AND ops involved. A sum tem Is equal to 1 when one or more of the literals In the term are 1, A sum tem Is equal to 0 only If each of the literals Is 0,
  5. Boolean Multiplication Boolean multiplication is equivalent to the AND operation A product term is produced by an AND operation with no OR ops involved. AB, AD, ABC,ÄBCD A product term Is equal to 1 only if each of the literals In the term Is 1. A product term Is equal to 0 when one or more of the literals are O,
  6. Laws & Rules of Boolean Algebra The basic laws of Boolean algebra: The commutative laws (nomjaätJil) The associative laws (nomäÖnn+J) The distributive laws (nnnojnjnu)
  7. Commutative Laws The commutative law of addition for two variables is written as: A+B = B+A The commutatzve law of multiplication for two variables is written as: AB = BA AB ABD.... B + A
  8. Associative Laws The associative law of addition for 3 variables is written as: A+(B+C) = (A+B)+C c B+C A+ (B+C) c The assoczatzve law of multiplication for 3 variables is written as: A(BC) = (AB)C
  9. Distributive Laws The distributive law is written for 3 variables as follows: X-A (B+C) x X=AB+AC
  10. Rules of Boolean Algebra 3.A,O- 10. A + AB = A, B, and C can represent a single variable or a combination of variables.
  11. DeMorgan's Theorems DeMorgan's theorems provide mathematical verification of: the equivalency of the N AND and negative-OR gates the equivalency of the NOR and negative-AND gates.
  12. The complement of two or more ANDed variables is equivalent to the OR of the complements of the Indlvldual varlables, The complement of two or more ORed varlables Is equivalent to the AND of the complements of the Individual variables, DeMorgan's Theorems XoY=R+V NAND Negative-OR Negative-AND Xo Y
  13. DeMorgan's Theorems (Exercises) Apply DeMofgan's theorems to the expressions: WoX0Y0Z
  14. DeMorgan's Theorems (Exercises) Apply DeMofgan's theorems to the expressions: ABC + DEF A+BC+D(E+F)
  15. Boolean Analysis of Logic Circuits Boolean algebra provides a concise way to express the operation of a logic circuit formed by a combination of logic gates so that the output can be determined for various combinations of Input values.
  16. Boolean Expression for a Logic Circuit To derive the Boolean expression for a given logic circuit, begin at the left-most inputs and work toward the final output, writing the expression for each gate, c D CD B+CD A (B+CD)
  17. Constructing a Truth Table for a Logic Circuit Once the Boolean expression for a given logic circuit has been determined, a truth table that shows the output for all possible values of the input variables can be developed. Let's take the previous circuit as the example: There are four variables, hence 16 (24) combinations of values are possible.
  18. Constructing a Truth Table for a Logic Circuit Evaluating the expression To evaluate the expression A(B+CD), first find the values of the variables that make the expression equal to 1 (using the rules for Boolean add & mult). In this case, the expression equals 1 only ifA=1 and B+CD=I because A(B+CD) = 101
  19. Constructing a Truth Table for a Logic Circuit Evaluating the expression (cont') Now, determine when B+CD term equals 1. The term B+CD=I if either or CD=I or if both B and CD equal 1 because B + CD = 1+0 = I B + CD = 0+1 = I B + CD = 1+1 = I The term CD=I only if C=l and D=/
  20. Constructing a Truth Table for a Logic Circuit Evaluating the expression (cont') Summary: • When A=l and B=l regardless of the values of C and D • When and C=l and D=l regardless of the value of B The expression A(B+CD)=O for all other value combinations of the variables.
  21. Constructing a Truth Table for a Logic Circuit Putting the results in truth table format When and B=l regardless of the values of C and D When A-I and and D=l regardless of the value of B INPUTS 1 1 1 1 1 1 OUTPUT A(B+CD) 1 1 1 1

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