Mathematical induction is a way of proving a given proposition about any well-organized set using mathematics. It's often used to prove or establish propositions that are expressed in terms of n, where n is a natural number. As mentioned below, the approach entails three stages to prove a statement, P(n):
- Verify if the statement is true for trivial cases like n = a, i.e. P(a) is true. [Base Case]
- Assume that the statement is true for n = k for some k ≥ a i.e. P(k) is true. [Inductive Hypothesis]
- If the truth of P(k) implies the truth of P(k + 1), then the statement P(n) is true for all n ≥ a.
The base step and the inductive step, together, prove that P(k) => P(k + 1) => P(k + 2) …. is true. Therefore, P(n) is true for all integers n ≥ a. Mathematical induction can be compared to dominoes falling. When a domino falls, it causes the following domino to fall in turn. The first domino topples the second, and the second topples the third, and so on. All of the dominoes will be knocked over in the end. However, there are several requirements that must be met:
- To begin the knocking process, the first domino must fall. This is the very first step.
- Any two adjacent dominoes must have the same space between them. Otherwise, a domino may fall without knocking down the next. The chain of reactions will then come to an end.
In this “Logic and Induction - Mathematical Foundation of Computer Science” you will learn about the following topics:
- Propositions and Truth functions
- Predicates and Quantification
- Propositional
- Predicate Logic
- Expressing statements in the language of Logic
- Deduction in Predicate Logic
- Elementary Step-wise Induction
- Complete Induction
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This article Logic and Induction - Mathematical Foundation of Computer Science is contributed by Namrata Chaudhary, a student of Lumbini Engineering College (LEC).
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