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Example. Express the Boolean function *F* = *x* + *y**z* as a sum of minterms.
Solution:** F** =

*x*+

*y*

*z*=

*x*+ (

*y*

*z*) AND (multiply) has a higher precedence than OR (add) =

*x*(

*y*+

*y*')(

*z*+

*z*') + (

*x*+

*x*')

*yz*expand 1st term by ANDing it with (

*y*+

*y*’)(

*z*+

*z*’), and 2nd term with (

*x*+

*x*’) =

*x y z*+

*x y z*' +

*x y*'

*z*+

*x y*'

*z*' +

*x y z*+

*x*'

*y z*=

*m*7 +

*m*6 +

*m*5 +

*m*4 +

*m*3 = Σ(3, 4, 5, 6, 7)

**sum of 1-minterms**

Example. Express the Boolean function *F* = *x* + *y**z* as a product of maxterms.
Solution: First, we need to convert the function into the product-of-OR terms by using the distributive law as
follows:
** F** =

*x*+

*y*

*z*=

*x*+ (

*y*

*z*) AND (multiply) has a higher precedence than OR (add) = (

*x*+

*y*) (

*x*+

*z*) use distributive law to change to product of OR terms = (

*x*+

*y*+

*z*

*z*') (

*x*+

*y*

*y*' +

*z*) expand 1st term by ORing it with

*z*

*z*', and 2nd term with

*y*

*y*' = (

*x*+

*y*+

*z*) (

*x*+

*y*+

*z*') (

*x*+

*y*+

*z*) (

*x*+

*y*' +

*z*) =

*M*0 •

*M*1 •

*M*2 = Π(0, 1, 2)

**product of 0-maxterms**

Example. Express *F* ' = (*x* + *y**z*)' as a sum of minterms.
Solution:
*F**'* = (*x* + *y**z*)' = (*x* + (*y**z*))' AND (multiply) has a higher precedence than OR (add)
= *x*' (*y*' + *z*') use dual or De Morgan’s Law
= (*x*' *y*') + (*x*' *z*') use distributive law to change to sum of AND terms
= *x*' *y*' (*z* + *z*') + *x*' (*y* + *y*') *z*' expand 1st term by ANDing it with (*z* + *z*'), and 2nd term with (*y* + *y*')
= *x*' *y*' *z* + *x*' *y*' *z*' + *x*' *y**z*' + *x*' *y*' *z*'
= *m*1 + *m*0 +* m*2
= Σ(0, 1, 2) **sum of 0-minterms**

Example. Express *F* ' = (*x* + *y**z*)' as a product of maxterms.
Solution:
** F'** = (

*x*+

*y*

*z*)' = (

*x*+ (

*y*

*z*))' AND (multiply) has a higher precedence than OR (add) =

*x*' (

*y*' +

*z*') use dual or De Morgan’s Law = (

*x*' +

*y*

*y*' +

*z*

*z*') (

*x*

*x*' +

*y*' +

*z*') expand 1st term by ORing it with

*y*

*y*' and

*z*

*z*', and 2nd term with

*x*

*x*' = (

*x*' +

*y*+

*z*) (

*x*' +

*y*+

*z*') (

*x*' +

*y*' +

*z*) (

*x*' +

*y*' +

*z*') (

*x*+

*y*' +

*z*') (

*x*' +

*y*' +

*z*') =

*M*4 •

*M*6 •

*M*5 •

*M*7 •

*M*3

= Π(3, 4, 5, 6, 7) **product of 1-maxterms***F* and *F'* are shown in the following truth table:

*x**y**z* Minterms Maxterms *F**F* '
0 0 0 *m*0=*x*' *y*' *z*' *M*0=*x* + *y* + *z* 0 1
0 0 1 *m*1=*x*' *y*' *z**M*1=*x* + *y* + *z*' 0 1
0 1 0 *m*2=*x*' *y z*' *M*2=*x* + *y*' + *z* 0 1
0 1 1 *m*3=*x*' *y z**M*3=*x* + *y*' + *z*' 1 0
1 0 0 *m*4=*x**y*' *z*' *M*4=*x*' + *y* + *z* 1 0
1 0 1 *m*5=*x**y*' *z**M*5=*x*' + *y* + *z*' 1 0
1 1 0 *m*6=*x**y**z*' *M*6=*x*' + *y*' + *z* 1 0
1 1 1 *m*7=*x**y**z**M*7=*x*' + *y*' + *z*' 1 0