BOOLEAN ALGEBRA • A Mathematical notation used to represent the function of the Digital circuit. • A notation that allows variables & constants to have only 2 possible values 0 & 1. • The Term Boolean Algebra honors a fascinating English mathematician; George Boole BOOLEAN FUNCTIONS -- An expression formed with Binary variables the two operators OR & AND & a UNARY operator not parenthesis & equal sign for the given variables the function can either be One or Zero.
BOOLEAN ALGEBRA • The Rules of Boolean Algebra: OR Operations (+) 0+0 = 0 A+0 = A 1+0 = 1 A+1 = 1 0+1 = 1 A+A = A 1+1 = 1 A+A' = 1 AND Operations (·) 0·0 = 0 A·0 = 0 1·0 = 0 A·1 = A 0·1 = 0 A·A = A 1·1 = 1 A·A' = 0 NOT Operations (') 0' = 1 A'' = A 1' = 0 Distributive Law A·(B+C) = (A·B) + (A·C) A+(B·C) = (A+B) · (A+C) Associative Law (A·B)·C = A·(B·C) = A·B·C (A+B)+C = A+(B+C) = A+B+C Precedence AB = A·B A·B+C = (A·B) + C A+B·C = A + (B·C) DeMorgan's Theorem (A·B)' = A' + B' (NAND) (A+B)' = A' · B' (NOR) Commutative Law A·B = B·A A+B = B+A
DE MORGAN’S THEOREM • A mathematician named De Morgan developed a pair of important rules regarding group terms complementation in Boolean algebra. • He states that inversion or negation of a Boolean expression may be accomplished by negating each variable and changing each AND to OR each OR to AND.
De Morgan’s Theorem Equations (xy)’ = x’ + y’ Is equivalent to (x+y )’ = x’y’
This statement can be further summarized to: • The First Theorem states that a NOR gate is Equivalent to a bubbled AND gate. • The second Theorem States that a NAND Gate is equivalent to a Bubbled OR Gate. • In these rule, two procedures are used to form a complement. • The (+) symbol are replaced with (·) symbol or absence of operation, and (·)symbols with (+) symbol. • Each term in the expression is then complemented. (A·B)' = A' + B' (NAND) (A+B)' = A' · B' (NOR)
NAND and NOR GATE IMPLEMENTATION • Another way of simplifying the circuit by using only the universal gates, NAND and NOR. • NAND and NOR are known to be Universal Gates because they can function as other gates
FORMS OF FUNCTIONS: • CANONICAL Forms • Boolean function expressed as Sum of Minterms or Product of Maxterms • MINTERM (denoted by lower case m) • A single variable or product of several variables which may or may not be complemented • A boolean expression containing ANDed Terms • Example: • A, A’B, A’B’C • F= A + A’B + A’B’C
FORMS OF EQUATIONS: • MAXTERM (denoted by the uppercase M) • A single variable or sum of several variables • A Boolean expression containing OR Terms. • Example: • A, A’+B, A’+B’ + C • (A) (A’+B) (A’+B’) (A’ + B’ + C) (A’ + B’ + C’)
MINTERM & MAXTERM Samples: • MINTERM: • F = A’BC + AB’C’ + A’ B’ C • F(A,B,C) =Σm3 + m4 + m1 • MAXTERM: • F = (A’+B+C) ( A+B’+C’ )(A’+ B’ +C) • F(A,B,C) =ΠM4 + M3 + M6
MINTERM & MAXTERM Samples: • Sample Problem:
SOP & POS Equations: • Standard Form of Equation: • The term that form the function may be one or any number or literal. • TWO TYPES: • Sum of Product (SOP) • A Boolean expression containing AND terms called PRODUCT of TERMS (one or more literals) • Ex: F=Y’ + XY+X’YZ’ • PRODUCT OF SUM (POS) • A Boolean expression containing OR terms called sum term. • Each term may have any number of literals • Ex: F=(X+Y) + (Y’+Z)(X’+Y’Z)
The Map Method K-MAPPING (Karnaugh) • Map a diagram made up of squares represents one • minterm • KARNAUGH MAP • A chart or grid containing boxes called cells; each which represents one minterm. • TYPES of MAP • 1.Two Variable Map • Consists of two variables • 2. Three Variable Map • A three variable map plotted in a map • 3. FOUR Variable Map
K-MAPPING (Karnaugh) • Steps: • Construct the K-Map & place 1’s in the squares corresponding to 1’s in truth table;place zeros in the other squares. • Examine the map for adjacent 1’s & loop those 1’s which are not adjacent to any other one’s. • Looping continue as there are pairs octet or quad that contains 1. You can still loop the one that is already looped if there are still other 1’s left. • Form the OR sum of all terms generated by each loop.
K-MAPPING (Karnaugh) The values of one variable appear across the top of the map, defining the column values, while the values of the other variable appear at the side, defining the values of the variable in each row. The Karnaugh map for the OR gate is completed by entering a '1' in each of the appropriate cells. Usually, you don't write in the '0's'. Within the map, adjacent cells containing 1's are grouped together in twos, fours, or eights. In this case, there is one horizontal and on vertical group of two. You indicate these groupings by drawing a circle round each one. The horizontal group corresponds to a B value of 1. In the left hand cell, A=0 and in the right hand cell, A=1. In other words, the value of A does not affect the outcome of the Boolean expression for these cells.
K-MAPPING (Karnaugh) K-MAP Types:
K-MAPPING (Karnaugh) • Don’t Care conditions: • It is represented by a letter (X) in the truth table instead of 0 or 1 • The X’s are treated either as 0 or 1’s whichever leads to a simpler circuit. • Five variable Map: • Consists of 5 variables .It uses 32 squares.
K-MAPPING (Karnaugh) • Example Don’t Care!