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ECE171

Winter2011

Homework3Solutions

1.  [10]Showthetruthtableforthefunction! =  ! + ! ∙ !.

2.  [20]Considerthefunction! = ! ∙ ! + ! ∙ !.

a)  Expandthefunctiontoitscanonicalorstandardsumofproducts(SOP)form.

b)  Expressthefunctioninitscanonicalorstandardproductofsums(POS)form.

c)  Expressthefunctionincompactmintermform.

d)  Expressthefunctionincompactmaxtermform.

Eachofthesequestionsismoreeasilyansweredifwefirstgeneratethetruthtable(thoughit’snot

essential).

a)  ! = ! ∙ ! ∙ ! +  ! ∙ ! ∙ ! + ! ∙ ! ∙ ! + ! ∙ ! ∙ !

b)  ! = (! + ! + !)∙(! + ! + !)

∙(! + ! + !)

∙(! + ! + !)

c)  ! = (1,3,6,7)

d)  ! = (0,2,4,5)

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8.  [15]UseaK-maptominimizethefunctionF w, x, y, z =   0,1,2,4,5,6,8,9,12,13,14

! = ! +! ∙ ! + ! ∙ ! 

9.  [15]UseaK-maptominimizethefunction

! A,B,C,D =  A ∙ B ∙ C+ B ∙ C ∙ D+ A ∙ B ∙ C ∙ D+ A ∙ B ∙ C

 

! = ! ∙ ! + ! ∙ ! +  ! ! ∙ !

10. [15]MinimizethefunctionaboveinPOSform.

Circling0sinsteadof1s,weget! =  ! ∙ ! + ! ∙ ! + ! ∙ !,applyingdeMorgan’sTheoremweget

! = !+ ! ∙ ! + ! ∙ (! + !)