Engin112-F07-L08-canonic

Engin112-F07-L08-canonic - Maciej Ciesielski Department of...

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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 Huntington’s postulates ¡ Boolean functions y Algebraic expression ¡ Today’s 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: yx’z OR y It is not humid AND ((its not raining AND it is hot ) OR it is raining ) y’ (x’z+x) ¡ Create a Boolean function that controls the fan. y F = yx’z + y’(x’z+x) y Other equivalent solutions : F = x’yz+x’y’z+xy’ OR F = x’z+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 : yx’z + y’(x’z+x) = yx’z+y’x’z+y’x = x’yz+x’y’z+xy’(z+z’) = x’yz+x’y’z+xy’z+xy’z’ 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 ¡ Huntington’s 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) x·1=x Post. 3: (a) x+y=y+x , (b) x·y=y·x Post. 4: (a) x(y+z) = xy+xz , (b) x+yz = (x+y)(x+z) Post. 5: (a) x+x’=1 , (b) x·x’=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) x·1=x Post. 3 : (a) x+y=y+x , (b) x·y=y·x Post. 4 : (a) x(y+z) = xy+xz , (b) x+yz = (x+y)(x+z) Post. 5 : (a) x+x’=1 , (b) x·x’=0 09/21/2007 Engin 112 - Intro to ECE 7 Proof of x·x=x ¡ Similar to previous proof ¡ Show that x·x = x . x·x = xx+0 by 2(a) = xx+xx’ by 5(b) = x(x+x’) by 4(a) = x·1 by 5(a) = x by 2(b) Q.E.D....
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This note was uploaded on 04/16/2008 for the course ENGIN 112 taught by Professor Ciesielski during the Spring '08 term at UMass (Amherst).

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Engin112-F07-L08-canonic - Maciej Ciesielski Department of...

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