Content
• Complement Function
• Standard Form
– Product of Sum (POS)
– Sum of Product (SOP)
– Minterm
– Maxterm
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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
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Boolean Function
• Example:
From TT we see that F3=F4
Can you prove it using Boolean Algebra?
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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’)’
= x’+y’+z
(DeMorgan)
(Involution)
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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’
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Standard Form
• There are two standard form:
Sum-of-Product (SOP) and Product-ofSum (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
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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’)
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Standard Form
• Every Boolean expression can be written
either in Sum-of-Product (SOP) expression
or Product-of- Sum (POS)
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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”
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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
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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
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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
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Canonical Form: Sum of Minterm (SOM)
• Produce TT: Example
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Canonical Form: Sum of Minterm (SOM)
• Produce Sum-of-Minterm by collecting
minterm for the function (where the answer
is 1)
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Canonical Form: Product of Maxterm (POM)
• Maxterm is “sum term”
• For Boolean function, maxterm for function
is term with answer 0
• Can state Boolean function in Product-ofMaxterm form
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Canonical Form: Product of Maxterm (POM)
• Example:
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Canonical Form: Product of Maxterm (POM)
• Why? Take F2 as example
• Complement function for F2 is
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Canonical Form: Product of Maxterm (POM)
• From the previous slide F2’=m0+m1+m2
Therefore:
• Notes: Complement of minterm = Maxterm
• Each Boolean function can be written in Sum-ofProduct and Product-of-Sum expression
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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)
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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-ofMaxterm for F’
– Maxterm list which is not in POM of F
E.g.
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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)
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Binary Function
• If n variable, therefore the are 2n 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
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Binary Function
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