1

Minterm and Maxterm Expressions

Definition: a minterm of n variables is a product of the variables

in which each appears exactly once in true or complemented form.

e.g.: minterms of 3 variables:

- Each minterm = 1 for only one combination of values of the variables, = 0 otherwise.

Definition: a maxterm of n variables is a sum of the variables

in which each appears exactly once in true or complemented form.

e.g.: 3 variables

- Each maxterm = 0 for only one combination of values of the variables, = 1 otherwise.

ABCA’BCAB’C

2n terms, n = no. variables

23 = 8

A+B+CA’+B+CA’+B’+C

2

All possible minterms and maxterms are obtained from the truth table:

e.g. recall from our votetaker:

F = ABC + ABC' + AB'C + A'BC

which can be written in terms of minterms as

F = m3 + m5 + m6 + m7

Row #

A B C Minterms Maxterms

0 0 0 0    

1 0 0 1    

2 0 1 0    

3 0 1 1    

4 1 0 0 AB'C'= A'+B+C=

5 1 0 1 AB'C= A'+B+C'=

6 1 1 0 ABC'= A'+B'+C=

7 1 1 1 ABC= A'+B'+C'=

e.g. 3 variables

How do we write minterm and maxterm expansions?

value =1 value = 0

A’B’C’ m0 A+B+C M0

A’B’C m1 A+B+C’ M1

A’BC’ m2 A+B’+C M2

A’BC m3 A+B’+C’ M3

m4 M4

m5 M5

m6 M6

m7 M7

111 110 101 011 7 6 5 3

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Minterms & Maxterms (continued)

which is abbreviated as F(A,B,C) =

- For each F = 1 row of truth table, only one mi = 1.

Therefore the minterm expansion is unique, i.e. there is a 1 to 1 correspondence between each minterm and each 1 in the truth table.

- Alternate form of F:

F = ( A + B + C ) ( A + B + C' ) ( A + B' + C ) ( A' + B + C )

or in terms of maxterms:

F =

or

F(A,B,C) =

- For each F = 0, only one Mi = 0.

Therefore maxterm expansion is unique.

m (3,5,6,7)

all others = 0

( 0 0 0 ) ( 0 0 1 )( 0 1 0 )( 1 0 0 ) 0 1 2 4

M0M1M2M4

M (0,1,2,4)

F = m (3,5,6,7) = M (0,1,2,4)

4

Minterms & Maxterms (continued)

(Note that we have been given a "simplified" expression, and we want to find the minterm expansion. This is moving in the opposite direction to what we did before, i.e. writing F from the truth table, and then simplifying).

-Note that if mi is present in minterm expansion, then Mi is not present in maxterm expansion , and conversely. - Note also that:

F' =- To convert from a general expression to a minterm or maxterm expansion, use:

a) truth tableor b) algebraic manipulation

e.g. Find the minterm expansion of: F = AB' + A'C

A B C F

0 0 0 0

0 0 1 0

0 1 0 0

0 1 1 1

1 0 0 0

1 0 1 1

1 1 0 1

1 1 1 1

m (0,1,2,4) = M (3,5,6,7)

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Minterms & Maxterms (continued)

b) Algebraically: use X + X' = 1 to introduce the missing variables in each term. Therefore F = AB' + A'C =

Solution of F = AB' + A'C

a) Using truth table:

Therefore F =

=

=

A B C F

0 0 0  

0 0 1  

0 1 0  

0 1 1  

1 0 0  

1 0 1  

1 1 0  

1 1 1  

Evaluate F

0

1

0

1

1

1

0

0

A’B’C+A’BC+AB’C’+AB’C

m1 + m3 + m4 + m5

m (1,3,4,5)

AB’(C+C’) + A’C(B+B’)

= AB’C + AB’C’ + A’BC + A’B’C

= m5 + m4 + m3 + m1

= m (1,3,4,5)

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Example

Find the maxterm expansion of F = ( A + B' ) ( A' + C )

b) Use XX' = 0 to introduce missing variables in each termTherefore F =

=

=

- Minterm and maxterm expansions are unique, therefore can prove equation F = G is valid by finding minterm or maxterm expansions

of both sides, and demonstrating the equality.

a) Truth Table F = (2,3,4,6)

X+YZ=(X+Y)(X+Z)(A+B’+C)(A+B’+C’)(A’+B+C)(A’+B’+C) 0 1 0 0 1 1 1 0 0 1 1 0 2 3 4 6(2,3,4,6)

(A + B’+ CC’)(A’ + C + BB’)x yz

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Incompletely Specified FunctionsIn some applications, certain combinations of inputs never occur, or the

output from certain combinations of inputs may be irrelevant.e.g.: The binary number 1010 - 1111 in BCD should never occur.In a truth table, the function F (at the output) is not important in such cases

and is said to be incompletely specified. We don't care what value (0 or 1) is assigned to F.

e.g.: A function of 3 variables; consider the truth table:

10 - 15

A B C F

0 0 0  

0 0 1  

0 1 0  

0 1 1  

1 0 0  

1 0 1  

1 1 0  

1 1 1  

1

X1

0

1

1

1

X2

1

F is incompletely Specified

X1X2 could be 00, 01, 10, 11

What values do we assign??

Ans. Choose values such that F is in simplest form.

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Incompletely Specified Functions(continued)

- When we expand F in minterm or maxterm, we must specify each x as 0 or 1. We should choose the values of x to produce the simplest form for F. Easiest to do this using a Karnaugh map (next topic).

- - In previous eg., simplest form for F is obtained by assigning

1 to 1st X1

0 to 2nd X2

yielding F =

after simplification

Formal minterm expansion would be written:

F =

A’B + BC

4 times with X1, X2 = 0, 1

m ( 0, 3, 7) + d ( 1, 6)Required terms + don’t care minterms