Boolean Function

1. 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

2. Boolean Function • Example: From TT we see that F3=F4 Can you prove it using Boolean Algebra? MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

3. 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: F1=xyz’ Complement F1’ = (xyz’)’ = x’+y’+(z’)’ (DeMorgan) = x’+y’+z (Involution) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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

5. 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

6. 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

7. 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

8. 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

9. Minterm & Maxterm • Generally, n variable will produce 2n 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 2n maxterm MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

10. Minterm & Maxterm • Minterm and maxterm for 2 variables each is signed with m0 to m3 and M0 to M1. Every minterm is the complement of suitable maxterm Example: m2=xy’ m2’=(xy’)’=x’+(y’)’=x’+y’=M2 MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

11. 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

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

13. 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

14. 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

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

16. Canonical Form: Product of Minterm (POM) • Why? Take F2 as example • Complement function for F2 is MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

17. Canonical Form: Product of Minterm (POM) • From the previous slide F2’=m0+m1+m2 Therefore: • Each Boolean function can be written in Sum-of-Product and Product-of-Sum expression MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

18. Canonical Form: Conversion SOPPOS • Sum-of-Minterm => Product-of-Maxterm • Change m to M • Insert minterm which is not in SOM • E.g. F1(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. F2(A,B,C)=  M(0,3,5,6)=  m(1,2,4,7) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

19. 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

20. Canonical Form: Conversion SOPPOS • Sum-of-Minterm for F => Product-of-Maxterm for F’ • Change m to M • E.g. F1(A,B,C)=m(3,4,5,6,7) F1’(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. F2(A,B,C)=M(0,1,2) F2’(A,B,C)=m(0,1,2) MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

21. Binary Function • If n variable, therefore the are 2n possible minterm • Each function can be expressed by Sum-of-Minterm, therefore there are 22 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 n MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

22. Binary Function MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR