Solutions to Exercises in .NET Embed Data Matrix 2d barcode in .NET Solutions to Exercises Solutions to Exercises.net Framework datamatrix 2d barcode integrating for .netusing barcode creator for visual .net control to generate, create data matrix 2d barcode image in visual .net applications."If]In the case that m+n is the smallest integer k such that kxn^m, the definition becomes scan data matrix ecc200 on .netUsing Barcode decoder for .net framework Control to read, scan read, scan image in .net framework applications.k ^ m+n = kxn ^ m . Bar Code barcode library with .netUsing Barcode scanner for VS .NET Control to read, scan read, scan image in VS .NET applications.Solution 7.4. For continued equivalences, pairs of repeated terms cancel each other out. So for continued equivalences, an even number of occurrences of the same term reduces to none, and an odd number of repeated terms reduces to one. D Solution 7.7. p v true { p y ( p = p) { py p =pv { true . Solution 7.10.Barcode writer in .netuse .net bar code integrated togenerate bar code for .netreflexivity of equivalence (5.4) } disjunction distributes over equivalence (7.5) } p reflexivity of equivalence (5.4) }Data Matrix encoder on c#.netusing barcode integrating for .net vs 2010 control to generate, create 2d data matrix barcode image in .net vs 2010 applications.{ golden rule, p,q := p,p } p = p= pyp { disjunction is idempotent } p =p = p { reflexivity of equivalence (5.4) }Control data matrix ecc200 data for .net gs1 datamatrix barcode data on .netSolution 7.12. = p/\(pyq) { golden rule: p,q := p,pvq }Control barcode data matrix data for vb datamatrix 2d barcode data in vb.netp = pyq = py (pyq)Incoporate barcode data matrix in .netusing vs .net crystal toencode data matrix ecc200 on asp.net web,windows applicationSolutions to Exercises barcode library on .netgenerate, create ean128 none in .net projectsassociativity and idempotence of disjunction }Paint barcode on .netuse .net vs 2010 bar code implementation tocreate barcode in .netreflexivity of equivalence (5.4) }Gs1128 encoding with .netgenerate, create gtin - 128 none with .net projectsp = pyq = pyq P Visual .net Crystal gtin - 14 printer on .netusing vs .net crystal togenerate ean / ucc - 14 with asp.net web,windows applicationSolution 7.13. Control bar code 39 data with .netto connect code-39 and uss code 39 data, size, image with .net barcode sdk(pyq) A(pyr)Control gs1 128 size on office excelto produce ean 128 and uss-128 data, size, image with office excel barcode sdkgolden rule: p,q := pyq,pyr Control upc-a supplement 5 data for vb.netto build upc code and upc a data, size, image with vb.net barcode sdkpyq = pyr = ( p v q ) v ( p y r )Control 2d data matrix barcode image with vb.netgenerate, create data matrix barcodes none with vb projectsassociativity, symmetry, idempotence of disjunction } disjunction distributes over equivalence } golden rule }Control upca data on microsoft word universal product code version a data for wordpyq = pyr = py qv r Asp.net Web Forms ecc200 printer for .netusing barcode maker for web.net control to generate, create data matrix barcode image in web.net applications.py (q = r = qvr)Print bar code for excel spreadsheetsgenerate, create barcode none for excel spreadsheets projectspy (q A T ) .Control ean13+2 image on office excelusing excel spreadsheets toproduce ean-13 supplement 5 with asp.net web,windows applicationSolution 7.14. Modus ponens:golden rule, p,q := p,p = q }p = p = q = py (p = q){ disjunction distributes over equivalence } p = p = q = pyp = pyq { simplification of continued equivalence, disjunction is idempotent } p = q = pvqpAq . golden rule }Solutions to Exercises De Morgan. The more complicated side is the right side (because it contains two negations rather than one).->p v ->q = = { { definition of negation } disjunction distributes over equivalence (applied twice) } p y q = false v q = p v false = false v false = { { false is unit of disjunction } rearranging terms (using symmetry and associativity of equivalence) } p = q = py q = false { = { golden rule } definition of negation } p /\q = false py q = q = p = false (p = false) v (q = false)De Morgan. Again we begin with the right side. -~>p A ~>q = { { golden rule } contrapositive applied to ->p = ->q rule just proved: ->p v ->q = -^(p A q) applied to third term } p = q = ^(p^q) = = { { { definition of negation } golden rule } definition of negation } p = q = pAq = false p v q = falseSolutions to Exercises DistributMty of conjunction over equivalence. p / \ ( q = r){ = golden rule, p,q := p,q = r }p = q = r = p v (q = r) { distributivity of disjunction over equivalence } p = q = r = pvq = pvr { rearrange terms (using symmetry and associativity of equivalence) and add p twice in order to head for the golden rule. }p = q = pvq = p = p = r = golden rule (twice once with p,q := p,q , once with p,q := p,r) }p Aq = p = p AT . We thus conclude that p A ( q = r) = pAq = p = p f \ r . Rearranging terms we get the required result. D Solution 7.15. In this solution, simplification of continued equivalences and disjunctions using the basic laws (symmetry, associativity, idempotence and constants) is not spelt out. We begin by deriving the equality between (a) and (b).( p y q ) A (qvr) A (r v p)= { { golden rule, p,q :- p vq , (q vr) A (r v p) = pvqv((qvr) A ( r v p } } distributivity of disjunction over conjunction and simplification } pvq = (qvr) A ( r v p ) = pvqvr = { golden rule, p,q := qvr ,rv p, simplification of continued disjunction } pvq = qvr = rvp = pv qvr = pv qvr = { simplification of continued equivalences } } pyq = (qvr) A (rvp)pvq = qvr = rvp . Solutions to Exercises Now the equality between (d) and (c) is obtained by replacing conjunction everywhere by disjunction and vice versa.(p Aq) v (qAr) v (r Ap)golden rule, p,q := p Aq , (qAr) v (r A p) } distributwity of conjunction over disjunction and simplification }p Aq ~ (qAr) v (r Ap) = pAqA((qAr) v (r A p ) )