Proof Techniques: Proofs by Induction

🌱 What Is Proof by Induction?

Imagine you set up a long line of dominoes on the floor.
If:

1. You push the first domino and it falls, and
2. Each domino is placed so that whenever one falls, it knocks down the next,

then the entire chain will fall — no matter how long it is.

Proof by induction works exactly the same way.
It has two parts: the base case and the induction step.

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🧩 1. Base Case

Here you show that the statement is true for the smallest value of n — often n = 0 or n = 1.

This is like saying:

“Let me check that the first domino does indeed fall.”

If the first step holds, you’re ready for the second part.

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🧩 2. Induction Step

Now you assume the statement is true for some value n = k.
This assumption is called the induction hypothesis.

Then you prove that the same statement must also be true for n = k + 1.

This part is like saying:

“If the domino at position k falls, I can guarantee that it will push the next one.”

Once you show this relationship, all steps become true automatically.

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🎨 Simple Diagram (Original ASCII)

                   PROOF BY INDUCTION
        ------------------------------------------------

                    Step 1: Base Case
                          n = 1
                            |
                            v
                  +------------------+
                  | Statement True   |
                  +------------------+

                            |
                            |  Step 2: Induction Step
                            v
       Assume true for n=k (Induction Hypothesis)
                            |
                            v
       +------------------+      +-------------------+
       | Statement True   | ---> | Statement True    |
       |     for n=k      |      |   for n=k+1       |
       +------------------+      +-------------------+

                            |
                            v
              Therefore, true for all natural n!

This picture shows the “chain reaction” idea behind induction.

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🌟 A Friendly Example (Very Easy and Intuitive)

Let’s look at something simple from computation:

Claim

A string made of n copies of the letter ‘a’ (like “aaa…a”) has length n.

Base Case: n = 1

The string “a” clearly has length 1.
✔ Base case holds.

Induction Step

Assume a string with k copies of ‘a’ has length k.
(This is our induction hypothesis.)

Now take a string with k + 1 copies of ‘a’.
That’s just the string of length k plus one more ‘a’.

So its length is:

k + 1.

✔ Induction step holds.

Therefore, by induction, the statement is true for all n.

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🧠 Why Induction Matters in Theory of Computation

Induction becomes essential whenever we deal with:

• Recursive definitions of languages
• Grammar derivations
• Structural proofs about automata
• Behavior of loops or repeated transitions
• Proofs about string lengths or pattern growth

Computers operate step by step, and induction mirrors this structure perfectly.

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