## CSCI 2824: Lecture 9In this lecture we will talk about the principle of mathematical induction and attempt to prove properties of numbers using induction.
## Mathematical InductionMathematical induction is a very common technique for proving properties of natural numbers (and other discrete structures such as sets, relations and trees that we will study very soon). Here is a video of falling dominoes: Click here. Let us imagine an infinitely long sequence of tiles arranged in a straight line (close enough to each other), and let us tip domino number 1. We wish to Base case: The first domino falls (we kicked it, so it falls). Inductive Step: Whenever a domino numbered n falls, then its successor numbered n+1, also falls. Therefore, we conclude that all the dominoes will fall.
The argument above is the crux of induction. To prove a property over all natural numbers , we may argue as follows: The property is true for (the first natural number). If the property is true for some natural number then it is true for natural number .
## Example-1Induction can be really useful to guess and prove closed forms of sequences. Consider a simple one: We have .
Therefore here is our
Proof
Proof is by induction on . Base Case: We verify that . So this works. Induction Hypothesis: For all , If then . Proof of induction hypothesis: Let be any given natural number such that . We seek to prove that .
Therefore, we have proved that for all by induction. ## Example-2
Proof
We prove this fact by induction just like we did for the dominoes.
Let be any given natural number such that holds. We wish to show We note that ## Example 3Let us try the sequence . First, we guess what the closed form could be: .
Proof
We prove by induction. The base case is for . Base Case: For , we verify .
Inductive Hypothesis: For all , If then .
## Weak Induction ProofsWe wish to prove a property for all natural numbers . I.e, . Proof by (weak) induction proceeds by establishing a base case: Base Case: Verify that holds. Induction Hypothesis: .
So far, we have been working with weak induction. We will now work with strong induction proofs. ## Strong Induction ProofsIn weak induction, we prove that the number satisfies by assuming that (its immediate predecessor) does. That may not always yield the simplest proof. ## Example 1Floor and Ceiling Functions
The function is also called the The function is also called the As examples, whereas . For negative numbers, it is a little counter intuitive: whereas . Consider the recurrence . Here is the result of performing the recurrence on a few values of . Some of you may recognize the pattern (it is rather important one for CS). Here is the We can now try proving it by induction. We will first use weak induction.
The theorem only applies to natural numbers . We handle this by simply allowing the base case of induction to start at . ## Failure of Proof by Weak Induction
This is not easy to prove and infact is strictly not true. This is because depends on and not on like our previous sequences. ## Proof by Strong InductionStrong induction is different from weak in the inductive hypothesis. Weak Induction: Assume prove . Strong Induction: Assume for all and prove .
Going back to dominoes, we assume in weak induction that the domino falls and prove that so does the . In strong induction, we assume that all dominoes numbered fall and prove that in that case the also falls.
Proof
Proof is by strong induction over .
(
For all If for all then . We will prove the strong induction hypothesis. We will split this into two cases based on being odd or not.
.
There is now an extra proof needed on the side that argues that whenever is even, . For completeness, here is the proof of the side claim.
Side Proof
This is an example of a proof by contradiction. Let us assume otherwise. I.e., and . In other words, we have Since is odd, we cannot have . Therefore, . As a result, there is a natural number between and . This is a contradiction. Therefore, it has to be the case that . |