Boolean Function Boolean function is an expression form

Boolean Function • Boolean function is an expression form containing binary variable, two-operator binary which is OR and AND, and operator NOT, sign ‘ and sign = • Answer is also in binary • We always use sign ‘. ’ for AND operator, ‘+’ for OR operator, ‘’’ or ‘ ’ for NOT operator. Sometimes we discard ‘. ’ sign if there is no contradiction MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 1

Boolean Function • Example: From TT we see that F 3=F 4 Can you prove it using Boolean Algebra? MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 2

Complement Function • Given function F, complement function for this function is F’, it is obtained by exchanging 1 with 0 on the output function F. • Example: F 1=xyz’ Complement F 1’ = (xyz’)’ = x’+y’+(z’)’ = x’+y’+z (De. Morgan) (Involution) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 3

Complement Function • Generally, complement function can be obtained using repeatedly De. Morgan Theorem (A+B+C+…. . +Z)’=A’. B’. C’. …. Z’ (A. B. C. …. . Z)’=A’+B’+C’+. …. +Z’ MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 4

Standard Form • There are two standard form: Sum-of-Product (SOP) and Product-of. Sum (POS) • Literals: Normal variable or in complement form. Example: x, x’, y, y’ • **Product: single literal or several literals with logical product (AND) Example: x, xyz’, A’B, AB MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 5

Standard Form • **Sum: single literal or several literals with logical sum (OR) Example: x, x+y+z’, A’+B, A+B • Sum-of-Product (SOP) expression: single product or several products with logical sum (OR) Example: x, x+yz’, xy’+x’yz, AB+A’B’ • Product-of- Sum (POS) expression: single sum or several sum with logical product (AND) Example: x, x. (y+z’), (x+y’)(x’+y+z), (A+B)(A’+B’) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 6

Standard Form • Every Boolean expression can be written either in Sum-of-Product (SOP) expression or Product-of- Sum (POS) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 7

Minterm & Maxterm • Consider two binary variable x, y • Every variable can exist as normal literal or in complement form (e. g. x, x’, &y, y’) • For two variables, there are four possible combinations with operator AND such as: x’y’, x’y, xy’, xy • This product is called minterm • Minterm for n variables is the number of “product of n literal from the different variables” MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 8

Minterm & Maxterm • Generally, n variable will produce 2 n minterm • With similar approach, maxterm for n variables is “sum of n literal from the different variables” Example: x’+y’, x’+y, x+y’, x+y • Generally, n variable will produce 2 n maxterm MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 9

Minterm & Maxterm • Minterm and maxterm for 2 variables each is signed with m 0 to m 3 and M 0 to M 1. Every minterm is the complement of suitable maxterm Example: m 2=xy’ m 2’=(xy’)’=x’+(y’)’=x’+y’=M 2 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 10

Canonical Form • What is canonical/normal form? – It is unique form to represent something • Minterm is “product term’ – Can state Boolean Function in Sum-of-Minterm MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 11

Canonical Form: Sum of Minterm (SOM) • Produce TT: Example MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 12

Canonical Form: Sum of Minterm (SOM) • Produce Sum-of-Minterm by collecting minterm for the function (where the answer is 1) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 13

Canonical Form: Product of Minterm (POM) • Maxterm is “sum term” • For Boolean function, maxterm for function is term with answer 0 • Can state Boolean function in Product-of. Maxterm form MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 14

Canonical Form: Product of Minterm (POM) • Example: MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 15

Canonical Form: Product of Minterm (POM) • Why? Take F 2 as example • Complement function for F 2 is MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 16

Canonical Form: Product of Minterm (POM) • From the previous slide F 2’=m 0+m 1+m 2 Therefore: • Each Boolean function can be written in Sum-of. Product and Product-of-Sum expression MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 17

Canonical Form: Conversion SOP POS • Sum-of-Minterm => Product-of-Maxterm – Change m to M – Insert minterm which is not in SOM – E. g. F 1(A, B, C)= m(3, 4, 5, 6, 7)= M(0, 1, 2) • Product-of-Maxterm => Sum-of-Minterm – Change M to m – Insert maxterm which is not in POM – E. g. F 2(A, B, C)= M(0, 3, 5, 6)= m(1, 2, 4, 7) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 18

Canonical Form: Conversion SOP POS • Sum-of-Minterm for F => Sum-of-Minterm for F’ – Minterm list which is not in SOM of F E. g. • Product-of-Maxterm for F => Product-of-Maxterm for F’ – Maxterm list which is not in POM of F E. g. MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 19

Canonical Form: Conversion SOP POS • Sum-of-Minterm for F => Product-of-Maxterm for F’ – Change m to M – E. g. F 1(A, B, C)= m(3, 4, 5, 6, 7) F 1’(A, B, C)= M(3, 4, 5, 6, 7) • Product-of-Maxterm for F=> Sum-of-Minterm for F’ – Change M to m – E. g. F 2(A, B, C)= M(0, 1, 2) F 2’(A, B, C)= m(0, 1, 2) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 20

Binary Function • If n variable, therefore the are 2 n possible minterm • Each function can be expressed by Sum-ofn 2 Minterm, therefore there are 2 different function • In two variable case, there is 22=4 possible minterm, and there is 24=16 different binary function • The 16 binary function is presented in the next slide MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 21

Binary Function MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR 22