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in g using barcode generation for software control to generate, create barcode code39 image in software applications. GS1 DataBar bar codes (n ). 1.4 Semantics of propositional logic = RHSn + (n + 1) by our induction hypothesis = = = = n (n+1) + (n + 1) 2 n (n +1) + 2 (n+1) arithmetic 2 2 (n+2) (n+1) arithmetic 2 ((n+1)+1) (n+1) arithmetic 2. = RHSn+1 . Since we succ Software Code-39 essfully showed the base case and the inductive step, we can use mathematical induction to infer that all natural numbers n have the property stated in the theorem above. 2 Actually, there are numerous variations of this principle.

For example, we can think of a version in which the base case is n = 0, which would then cover all natural numbers including 0. Some statements hold only for all natural numbers, say, greater than 3. So you would have to deal with a base case 4, but keep the version of the inductive step (see the exercises for such an example).

The use of mathematical induction typically suceeds on properties M (n) that involve inductive de nitions (e.g. the de nition of k l with l 0).

Sentence (3) on page 2 suggests there may be true properties M (n) for which mathematical induction won t work. Course-of-values induction. There is a variant of mathematical induction in which the induction hypothesis for proving M (n + 1) is not just M (n), but the conjunction M (1) M (2) M (n).

In that variant, called courseof-values induction, there doesn t have to be an explicit base case at all everything can be done in the inductive step. How can this work without a base case The answer is that the base case is implicitly included in the inductive step. Consider the case n = 3: the inductive-step instance is M (1) M (2) M (3) M (4).

Now consider n = 1: the inductive-step instance is M (1) M (2). What about the case when n equals 0 In this case, there are zero formulas on the left of the , so we have to prove M (1) from nothing at all. The inductive-step instance is simply the obligation to show M (1).

You might nd it useful to modify Figure 1.9 for course-of-values induction. Having said that the base case is implicit in course-of-values induction, it frequently turns out that it still demands special attention when you get inside trying to prove the inductive case.

We will see precisely this in the two applications of course-of-values induction in the following pages.. 1 Propositional logic Figure 1.10. A parse tree with height 5. In computer science, we often deal with nite structures of some kind, data structures, programs, les etc. Often we need to show that every instance of such a structure has a certain property. For example, the well-formed formulas of De nition 1.

27 have the property that the number of ( brackets in a particular formula equals its number of ) brackets. We can use mathematical induction on the domain of natural numbers to prove this. In order to succeed, we somehow need to connect well-formed formulas to natural numbers.

De nition 1.32 Given a well-formed formula , we de ne its height to be 1 plus the length of the longest path of its parse tree. For example, consider the well-formed formulas in Figures 1.

3, 1.4 and 1.10.

Their heights are 5, 6 and 5, respectively. In Figure 1.3, the longest path goes from to to to to r, a path of length 4, so the height is 4 + 1 = 5.

Note that the height of atoms is 1 + 0 = 1. Since every well-formed formula has nite height, we can show statements about all well-formed formulas by mathematical induction on their height. This trick is most often called structural induction, an important reasoning technique in computer science.

Using the notion of the height of a parse tree, we realise that structural induction is just a special case of course-of-values induction..
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