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Unformatted text preview: Maciej Ciesielski Department of Electrical and Computer Engineering 09/21/2007 Engin112 Lectures 8,10 Boolean Functions: Canonical Forms 09/21/2007 Engin 112  Intro to ECE 2 Recap from Last Lecture Boolean algebra y Algebra axioms, postulates y Huntingtons postulates Boolean functions y Algebraic expression Todays lecture y Boolean Theorems y Comparison of Boolean functions y Canonical and standard forms 09/21/2007 Engin 112  Intro to ECE 3 Design Problem Design the control logic for a fan in a greenhouse. The logic must sense three environmental conditions: 1. It is raining outside (variable x ) 2. It is humid inside (variable y ) 3. It is hot inside (variable z ) The fan should turn on when either condition is met y It is humid AND it is not raining AND it is hot: yxz OR y It is not humid AND ((its not raining AND it is hot ) OR it is raining ) y (xz+x) Create a Boolean function that controls the fan. y F = yxz + y(xz+x) y Other equivalent solutions : F = xyz+xyz+xy OR F = xz+xy 09/21/2007 Engin 112  Intro to ECE 4 Boolean Function Representations Greenhouse example function can be expressed as: A nalytically, as sum of minterms : yxz + y(xz+x) = yxz+yxz+yx = xyz+xyz+xy(z+z) = xyz+xyz+xyz+xyz 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 F z y x As a logic network As a truth table How to derive different solutions ? How to evaluate these solutions ? 09/21/2007 Engin 112  Intro to ECE 5 Boolean Theorems Huntingtons postulates define some rules Need more rules to modify algebraic expressions y Theorems that are derived from postulates What is a theorem? y A formula or statement that is derived from postulates (or other proven theorems) Basic theorems of Boolean algebra y Theorem 1 (a): x + x = x (b): x x = x y Looks straightforward, but needs to be proven ! Post. 1: closure Post. 2: (a) x+0=x , (b) x1=x Post. 3: (a) x+y=y+x , (b) xy=yx Post. 4: (a) x(y+z) = xy+xz , (b) x+yz = (x+y)(x+z) Post. 5: (a) x+x=1 , (b) xx=0 09/21/2007 Engin 112  Intro to ECE 6 Proof of x+x=x We can only use Huntington postulates: Show that x+x=x . x+x = (x+x)1 by 2(b) = (x+x)(x+x) by 5(a) = x+xx by 4(b) = x+0 by 5(b) = x by 2(a) Q.E.D. We can now use Theorem 1(a) in future proofs Huntington postulates : Post. 2 : (a) x+0=x , (b) x1=x Post. 3 : (a) x+y=y+x , (b) xy=yx Post. 4 : (a) x(y+z) = xy+xz , (b) x+yz = (x+y)(x+z) Post. 5 : (a) x+x=1 , (b) xx=0 09/21/2007 Engin 112  Intro to ECE 7 Proof of xx=x Similar to previous proof Show that xx = x . xx = xx+0 by 2(a) = xx+xx by 5(b) = x(x+x) by 4(a) = x1 by 5(a) = x by 2(b) Q.E.D....
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 Spring '08
 ciesielski

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