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Solution 1.1. For n = 5, the number of portions is 16. This suggests that the number of portions for arbitrary n is 2n~l. However, for n = 0, it does not make sense to say that there are 2 W ~ 1 portions (even though cutting the cake as stated does make sense). The conclusion might then be that the conjecture only holds for n at least 1. However, for n = 6, the number of portions is 31 (see Figure B.I)! Note that n - 6 is the first case in which the points are not allowed to be placed at equal distances around the perimeter.) n



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One option for inserting elements into a dictionary is to use the Add() method, passing both the key and the value, as shown in Listing 165

and help files, with dynamic content such a customized tools, tailored processes, templates, wizards, and tests, all integrated into Visual Studio for producing a particular type of solution DSL Tools form an important part of this initiative

using System; using SystemCollectionsGeneric; class Program { static void Main() { Dictionary<Guid,string> dictionary = new Dictionary<Guid, string>(); Guid key = GuidNewGuid(); dictionaryAdd(key, "object"); } }

Solution 2.1. (a) Hoare assumes that the pack contains 100 cards. The weakest assumption is that the pack contains at least 20 cards. (b) The number of cards in the top-left heap is always at most 20. The number of cards in the top-right heap is always at most 80.





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Listing 165 inserts the string "object" using a Guid as its key If an element with the same key has already been added, an exception is thrown An alternative is to use the indexer, as shown in Listing 166

Before talking about graphical DSLs, let s look brie y at textual DSLs We ll see how Domain-Speci c Development involves a particular way of thinking about a problem, and we ll look at how to implement this approach using textual languages Imagine that we are designing a graphical modeling tool and have the problem of de ning a set of shapes that will be displayed on a screen to represent the various concepts that can be depicted by the tool One way we might do this would be to invent a new textual language for de ning the various shapes A fragment of this language might look like this:

using System; using SystemCollectionsGeneric; class Program { static void Main() { Dictionary<Guid, string> dictionary = new Dictionary<Guid, string>(); Guid key = GuidNewGuid(); dictionary[key] = "object"; dictionary[key] = "byte"; } }

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The first thing to observe in Listing 166 is that the index operator does not require an integer Instead, the index data type is specified by the first type parameter, TKey, when declaring a Dictionary<TKey, TValue>

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A variation on the principle of mathematical induction is the so-called strong induction principle3. Formally, in a proof by strong induction, there is no separate basis step and the induction step uses the assumption ( V k : 0 ^ in order to prove P.n. The precise formulation is as follows. Principle of Strong Mathematical Induction. Let P be a predicate on the natural numbers. Then, with dummy n ranging over the natural numbers, (Vn::P.n) = < V n :: P.n <= {Vfc:0^ The word 'strong' is used because the induction hypothesis when proving P.(n+l) is P.O A P.I A ... A P.n, which is stronger than the hypothesis P.n used in simple mathematical induction. The absence of a base case is sometimes illusory; occasionally, a special case needs to be made for n = 0 in the induction step as the induction hypothesis is (V k: 0 < k < 0: P.k), i.e. true, which is just the same as in simple induction.

Define AnnotationShape Rectangle Width=15 Height=03 FillColor=khaki OutlineColor=brown Decorator Comment Position="Center" End Comment End AnnotationShape

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