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

2
Boolean Function

• Example
• From TT we see that F3F4
• Can you prove it using Boolean Algebra?

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 F1xyz
• Complement
• F1 (xyz)
• xy(z) (DeMorgan)
• xyz (Involution)

4
Complement Function

• Generally, complement function can be obtained
  using repeatedly DeMorgan Theorem
• (ABC..Z)A.B.C..Z
• (A.B.C...Z)ABC..Z

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, AB, AB

6
Standard Form

• Sum single literal or several literals with
  logical sum (OR)
• Example x, xyz, AB, AB
• Sum-of-Product (SOP) expression single product
  or several products with logical sum (OR)
• Example x, xyz,xyxyz, ABAB
• Product-of- Sum (POS) expressionsingle sum or
  several sum with logical product (AND)
• Example x, x.(yz),(xy)(xyz), (AB)(AB)

7
Standard Form

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

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
  xy,xy,xy,xy
• This product is called minterm
• Minterm for n variables is the number of product
  of n literal from the different variables

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 xy, xy, xy, xy
• Generally, n variable will produce 2n maxterm

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 m2xy
• m2(xy)x(y)xyM2

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

12
Canonical Form Sum of Minterm (SOM)

• Produce TT Example

13
Canonical Form Sum of Minterm (SOM)

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

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

15
Canonical Form Product of Minterm (POM)

• Example

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Canonical Form Product of Minterm (POM)

• Why? Take F2 as example
• Complement function for F2 is

17
Canonical Form Product of Minterm (POM)

• From the previous slide F2m0m1m2
• Therefore
• Each Boolean function can be written in
  Sum-of-Product and Product-of-Sum expression

18
Canonical Form Conversion SOP?POS

• Sum-of-Minterm gt 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 gt 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)

19
Canonical Form Conversion SOP?POS

• Sum-of-Minterm for F gt Sum-of-Minterm for F
• Minterm list which is not in SOM of F
• E.g.
• Product-of-Maxterm for F gt Product-of-Maxterm
  for F
• Maxterm list which is not in POM of F
• E.g.

20
Canonical Form Conversion SOP?POS

• Sum-of-Minterm for F gt 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 Fgt 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)

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 224 possible
  minterm, and there is 2416 different binary
  function
• The 16 binary function is presented in the next
  slide

n
22
Binary Function