Is mathematical induction is about proving the expression or its validity over set of value assuming expression is correct.

Question

Is mathematical induction is about proving the expression or is it about proving that expression is valid over a set of value(natural number), assuming expression is correct?

Answer 1

Can you explain what you mean a little more? Induction can be described by a sort of domino effect. We prove that if we can knock over the first domino, and if one domino knocks over the succeeding domino, then we have knocked over all the dominoes. This can be used to prove a range of things that require the natural numbers. It can be something from a closed form expression of a recursive sequence, to facts about generators and relations of some algebra. Its use is very diverse.

Answer 2

Mathematical induction is a principle that can be used to prove a statement is true for all natural numbers (or all natural numbers past some initial starting point.)

The steps for induction are as follows. Prove that the statement is true for an initial value (usually $n=1$). This is known as the base case. Then we prove that if the statement is true for $n$ then it is true for $n+1$. This proves the statement for all $n$.

For instance, if you wanted to prove a statement like,

For all $n \in \mathbb{N}$ prove that $1+2+3+4 ... +n = \frac{n(n+1)}{2}$. We can verify that this is true for $n=1$ or $n=2$ or $n=3$, but how can we prove this is true for all possible $n$? This is where we can use induction.

We note that for $n=1$, $1 = \frac{1 \cdot (2)}{2}$ so the formula holds for $n = 1$.

Now suppose that for some $n \in \mathbb{N}$ that $1+2+3...+n = \frac{n(n+1)}{2}$. Then $1+2+3...+n + (n+1) = \frac{n(n+1)}{2}+ (n+1) = \frac{(n+1)(n+2)}{2}$. So the formula held for $n+1$! Thus by the principle of mathematical induction, the formula holds for all $n$.

I hope that helps!