Truth Table Relationship between a function and variable Logic Diagram

# Truth table relationship between a function and

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Truth Table: Relationship between a function and variable Logic Diagram: Algebraic Expression Comp Architecture - Digital Computer 13
Purpose of Boolean Algebra To facilitate the analysis and design of digital circuit F= AB’ + C’D + AB’ + C’D = x + x (let x= AB’ + C’D) = x = AB’ + C’D Boolean function = Algebraic form = convenient tool Truth table (relationship between binary variables ) Algebraic form Logic diagram (input-output relationship ) Algebraic form Find simpler circuits for the same function : by using Boolean algebra rules Comp Architecture - Digital Computer 14
Graphic symbols for NOR & NAND gate Comp Architecture - Digital Computer 15
Rule in Boolean algebra Variable used can have only two values. Binary 1 for HIGH and Binary 0 for LOW. Complement of a variable is represented by an over bar (-). Thus complement of variable B is represented as. Thus if B = 0 then = 1 and B = 1 then = 0. ORing of the variables is represented by a plus (+) sign between them. For example ORing of A, B, C is represented as A + B + C. Comp Architecture - Digital Computer 16
Cont… Logical ANDing of the two or more variable is represented by writing a dot between them such as A.B.C. Sometime the dot may be omitted like ABC. Comp Architecture - Digital Computer 17
Boolean Laws COMMUTATIVE LAW A.B=B.A OR A+B=B+A ASSOCIATIVE LAW (A.B).C= A.(B.C) OR (A+B)+C= A+(B+C) DISTRIBUTIVE LAW A.(B+C)=(A.B)+(A.C) AND LAW A.0=0 A.1=1 A.A=A A.A=O Comp Architecture - Digital Computer 18
Cont… Comp Architecture - Digital Computer 19 OR LAW A.0=0 A.1=A A.A=A A.A=0 INVERSION LAW The inversion law states that double inversion of a variable result in the original variable itself. A ’ ’=A
Truth Table Formation Comp Architecture - Digital Computer 20 A truth table represents a table having all combinations of inputs and their corresponding result
Map Simplification Karnaugh Map(K-Map) Map method for simplifying Boolean expressions Minterm / Maxterm Minterm : n variables product ( x=1, x’=0) Maxterm : n variables sum (x=0, x’=1) Comp Architecture - Digital Computer 21
F(x, y, z) = Σ(1,4,5,6,7) Comp Architecture - Digital Computer 22
Adjacent Square Number of square = 2n (2, 4, 8, ….) The squares at the extreme ends of the same horizontal row are to be considered adjacent The same applies to the top and bottom squares of a column The four corner squares of a map must be considered to be adjacent Comp Architecture - Digital Computer 23
Adjacent Squares Comp Architecture - Digital Computer 24
Example 1,2 F(A,B,C) = Σ(3,4,6,7) F=AC’ + BC F(A,B,C) = Σ(0,2,4,5,6) F=C’ + AB’ Comp Architecture - Digital Computer 25
Example 3 F(A,B,C,D) = Σ(0,1,2,6,8,9,10) F=B’D’ + B’C’ + A’CD’ Comp Architecture - Digital Computer 26
LOGIC GATE Comp Architecture - Digital Computer 27

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• Boolean Algebra, Logic gate, Combinational Circuits,  Examples of the digital systems