1MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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

2MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Boolean Function

• Example:

From TT we see that F3=F4

Can you prove it using Boolean Algebra?

3MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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)

4MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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’

5MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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

6MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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’)

7MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Standard Form

• Every Boolean expression can be written either in Sum-of-Product (SOP) expression or Product-of- Sum (POS)

8MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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”

9MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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

10MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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 maxtermExample: m2=xy’m2’=(xy’)’=x’+(y’)’=x’+y’=M2

11MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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

12MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Canonical Form: Sum of Minterm (SOM)

• Produce TT: Example

13MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Canonical Form: Sum of Minterm (SOM)

• Produce Sum-of-Minterm by collecting minterm for the function (where the answer is 1)

14MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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

15MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Canonical Form: Product of Minterm (POM)

• Example:

16MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Canonical Form: Product of Minterm (POM)

• Why? Take F2 as example

• Complement function for F2 is

17MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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

18MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Canonical Form: Conversion SOPPOS

• 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)

19MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Canonical Form: Conversion SOPPOS

• 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.

20MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Canonical Form: Conversion SOPPOS

• 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)

21MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

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

22MOHD. YAMANI IDRIS/ NOORZAILY MOHAMED NOOR

Binary Function