Boolean Algebra. Boolean Algebras

Transcription

1 Boolen Algebr Boolen Algebrs A Boolen lgebr is set B of vlues together with: - two binry opertions, commonly denoted by + nd, - unry opertion, usully denoted by or ~ or, - two elements usully clled zero nd one, such tht for every element of B: nd In ddition, certin ioms must be stisfied: - closure properties for both binry opertions nd the unry opertion - ssocitivity of ech binry opertion over the other, - commuttivity of ech ech binry opertion, - distributivity of ech binry opertion over the other, - bsorption rules, - eistence of complements with respect to ech binry opertion We will ssume tht hs higher precedence thn +; however, this is not generl rule for ll Boolen lgebrs. CS@VT September McQuin, Feng & Ribbens

2 Aioms of Boolen Algebr Associtive Lws: for ll, b nd c in B, Boolen Algebrs 2 ( b) c ( b c) ( b) c ( b c) Commuttive Lws: for ll nd b in B, b b b b Distributive Lws: for ll, b nd c in B, ( b c) ( b) ( c) ( b c) ( b) ( c) Absorption Lws: for ll, b nd c in B, ( b) ( b) Eistence of Complements: for ll in B, there eists n element ā in B such tht CS@VT September McQuin, Feng & Ribbens

3 Emples of Boolen Algebrs Boolen Algebrs 3 The clssic emple is B = {true, flse} with the opertions AND, OR nd NOT. An isomorphic emple is B = {, } with the opertions +, nd ~ defined by: b + b b ~ Given set S, the power set of S, P(S) is Boolen lgebr under the opertions union, intersection nd reltive complement. Other, interesting emples eist CS@VT September McQuin, Feng & Ribbens

4 More Properties It's lso possible to derive some dditionl fcts, including: - the elements nd re unique - the complement of n element is unique - nd re complements of ech other Boolen Algebrs 4 CS@VT September McQuin, Feng & Ribbens

5 DeMorgn's Lws & More Boolen Algebrs 5 DeMorgn's Lws re useful theorems tht cn be derived from the fundmentl properties of Boolen lgebr. For ll nd b in B, b b b b Of course, there s lso double-negtion lw: And there re idempotency lws: Boundedness properties: CS@VT September McQuin, Feng & Ribbens

6 Logic Epressions nd Equtions Boolen Algebrs 6 A logic epression is defined in terms of the three bsic Boolen opertors nd vribles which my tke on the vlues nd. For emple: z : y y y : ( 2) ( 2 3) ( 2 3) A logic eqution is n ssertion tht two logic equtions re equl, where equl mens tht the vlues of the two epressions re the sme for ll possible ssignments of vlues to their vribles. For emple: y y y y Of course, equtions my be true or flse. Wht bout the one bove? CS@VT September McQuin, Feng & Ribbens

7 September McQuin, Feng & Ribbens Boolen Algebrs 7 Why do they cll it "lgebr"? A Boolen epression cn often be usefully trnsformed by using the theorems nd properties stted erlier: y y y y y y y y This is reltively simple emple of reduction. Try showing the following epressions re equl: z y z y

8 Tutologies, Contrdictions & Stisfibles Boolen Algebrs 8 A tutology is Boolen epression tht evlutes to true () for ll possible vlues of its vribles. b b b b A contrdiction is Boolen epression tht evlutes to flse () for ll possible vlues of its vribles. A Boolen epression is stisfible if there is t lest one ssignment of vlues to its vribles for which the epression evlutes to true (). b b CS@VT September McQuin, Feng & Ribbens

9 Truth Tbles Boolen Algebrs 9 A Boolen epression my be nlyzed by creting tble tht shows the vlue of the epression for ll possible ssignments of vlues to its vribles: b b b b b CS@VT September McQuin, Feng & Ribbens

10 September McQuin, Feng & Ribbens Boolen Algebrs Proving Equtions with Truth Tbles Boolen equtions my be proved using truth tbles (dull nd mechnicl): c b c b + c b c b c b

11 Proving Equtions Algebriclly Boolen Algebrs Boolen equtions my be proved using truth tbles, which is dull nd boring, or using the lgebric properties: B, bsorption, with b lw of complements B, bsorption, with b lw of complements Note the dulity CS@VT September McQuin, Feng & Ribbens

12 Proving Equtions Algebriclly Boolen Algebrs2 B, bsorption, with b ( ) lw of complements bsorption, with b = CS@VT September McQuin, Feng & Ribbens

13 Sum-of-Products Form Boolen Algebrs3 A Boolen epression is sid to be in sum-of-products form if it is epressed s sum of terms, ech of which is product of vribles nd/or their complements: b b It's reltively esy to see tht every Boolen epression cn be written in this form. Why? The summnds in the sum-of-products form re clled minterms. - ech minterm contins ech vribles, or its complement, ectly once - ech minterm is unique, nd therefore so is the representtion (side from order) CS@VT September McQuin, Feng & Ribbens

14 Emple Boolen Algebrs4 Given truth tble for Boolen function, construction of the sum-of-products representtion is trivil: - for ech row in which the function vlue is, form product term involving ll the vribles, tking the vrible if its vlue is nd the complement if the vrible's vlue is - tke the sum of ll such product terms y z F y z y z y z y z F y zy zy zy z CS@VT September McQuin, Feng & Ribbens

15 Product-of-Sums Form Boolen Algebrs5 A Boolen epression is sid to be in product-of-sums form if it is epressed s product of terms, ech of which is sum of vribles: b b Every Boolen epression cn lso be written in this form, s product of mterms. Fcts similr to the sum-of-products form cn lso be sserted here. The product-of-sums form cn be derived by epressing the complement of the epression in sum-of-products form, nd then complementing. CS@VT September McQuin, Feng & Ribbens

16 Emple Boolen Algebrs6 Given truth tble for Boolen function, construction of the product-of-sums representtion is trivil: - for ech row in which the function vlue is, form product term involving ll the vribles, tking the vrible if its vlue is nd the complement if the vrible's vlue is - tke the sum of ll such product terms; then complement the result y z F y z y z y z y z F y zy zy zy z F yzyzyzyz ( yz) ( yz) ( yz) ( yz) CS@VT September McQuin, Feng & Ribbens

17 Boolen Functions Boolen Algebrs7 A Boolen function tkes n inputs from the elements of Boolen lgebr nd produces single vlue lso n element of tht Boolen lgebr. For emple, here re ll possible 2-input Boolen functions on the set {, }: A B zero nd A B or or A B nor eq B' A' nnd one CS@VT September McQuin, Feng & Ribbens

18 Universlity Boolen Algebrs8 Any Boolen function cn be epressed using: - only AND, OR nd NOT - only AND nd NOT - only OR nd NOT - only AND nd XOR - only NAND - only NOR The first ssertion should be entirely obvious. The remining ones re obvious if you consider how to represent ech of the functions in the first set using only the relevnt functions in the relevnt set. CS@VT September McQuin, Feng & Ribbens