APPLICATION OF BOOLEAN ALGEBRA(MINTERMS & MAXTERMS EXPANSIONS)

PRESENTED BY :NOUSHEEN ARSHAD, TAYYABA SHAHZAD, HAMMAD KHAN & HASSAN AYAZ KIANI

: MINTERMS :

• A MINTERM (PRODUCT) IS A COMBINATION OF VARIABLES :• IT HAS A VALUE OF 1 FOR ONLY ONE INPUT COMBINATION.• IT IS 0 FOR ALL THE OTHER COMBINATIONS OF VARIABLES.

• A MINTERM OF N VARIABLES IS THE PRODUCT OF N LITERALS IN WHICH EACH VARIABLE APPEARS ONCE IN EITHER A COMPLEMENTED OR UNCOMPLEMENTED FORM.

• FOR EXAMPLE: IF A FUNCTION OF 3 VARIABLES X,Y AND Z PRODUCES AS 1 OUTPUT FOR XYZ=010, 100, 111, THEN IT CAN BE WRITTEN AS F= X'YZ'+ XY'Z'+ XYZ.

:

MINTERMS

EXPANSION

:

• IF A MINTERM HAS A SINGLE 1 AND THE REMAINING CELLS AS 0'S, IT WOULD APPEAR TO COVER A MINIMUM AREA OF 1'S.

• IT IS STANDARD SUM-OF-PRODUCTS (SOP).

• F= A'BC+AB'C'+AB'C+ABC HAS FOUR MINTERMS.

• THIS FUNCTION WILL BE WRITTEN AS

F= Σ M(3,4,5,7).

: EXAMPLE :

: MAXTERMS :

• A MAX (SUM) TERM IS ALSO A UNIQUE COMBINATION OF VARIABLES :• HOWEVER, IT IS OPPOSITE OF MINTERMS.• IT HAS A VALUE OF 0 FOR ONLY ONE INPUT COMBINATION.• IT IS 1 FOR ALL THE OTHER COMBINATIONS OF VARIABLES.• THAT IS WHY, IT IS CALLED MAX (SUM) TERMS.FOR EXAMPLE: THE SAME FUNCTION OF 3 VARIABLES X,Y AND Z PRODUCES AS 0 OUTPUT FOR XYZ=000, 001, 011, 101, 110, THEN F= (X+Y+Z).(X+Y+Z').(X+Y'+Z').(X'+Y+Z').(X'+Y'+Z)

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MAXTERMS

EXPANSION

:

• IT IS A STANDARD PRODUCT OF SUM (POS).

• WE CAN ALSO TREAT AN ORDER OF VARIABLES TO REPRESENT AN INTEGER .

• THE FUNCTION CAN ALSO BE WRITTEN AS F(X,Y,Z) = Π M(0,1,3,5,6).

: EXAMPLE :

: CONVERSION OF ENGLISH SENTENCES TO BOOLEAN EQUATIONS :

• FOR SIMPLE PROBLEMS, GO DIRECTLY FROM A WORD DESCRIPTION OF THE DESIRED CIRCUIT BEHAVIOR TO AN BOOLEAN EXPRESSION.

• FOR EXAMPLE: • THE ALARM WILL RING (Z) IFF• THE ALARM IS TURNED ON (A), AND• THE DOOR IS NOT CLOSED (B'), OR• OR, IT IS AFTER 6 P.M (C), AND• THE WINDOW IS NOT CLOSED (D').• Z = AB'+CD'.

: COMBINATIONAL LOGIC DESIGN USING A TRUTH TABLE :

• EXAMPLE NO 01:• THE CANONICAL MINTERM (SOP) FORM F= A'B'+A'B.• THE CANONICAL MAXTERM (POS) FORM F= (A'+B).(A'+B').

: EXAMPLE NO 02 :

• MAXTERM EXPANSION:• F(A,B,C) = (A+B+C).(A+B+C').(A+B'+C) = M0 + M1+ M2. = Π M(0,1,2).

• MINTERM EXPANSION:• F(A,B,C) =

A'BC+AB'C'+AB'C+ABC'+ABC. = M3+M4+M5+M6+M7. = Σ M(3,4,5,6,7).

: DESIGN OF BINARY ADDERS :

• DESIGN SIMPLE BINARY ADDERS THAT ADDS TWO-BIT BINARY NUMBERS (N1, N2) AND PRODUCES 3-BIT SUM (N3).

: BINARY ADDER :

• 1 BIT BINARY ADDER...

• INPUTS: A,B, CARRY-IN.

• OUTPUT: SUM, CARRY-OUT.

• SUM = A'B'CIN + A'BCIN' + AB'CIN' + ABCIN.

• COUT = A'BCIN + AB'CIN + ABCIN' + ABCIN.

• BOTH SUM AND COUT CAN BE MINIMIZED.