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Suppose you have to prove the solution to the following recurrence by Induction,

$$ T(n)= \begin{cases} \Theta(1), & n=1 \\ 2 T(\lfloor n/2 \rfloor)+\Theta(n), & n>1 \end{cases} $$ Here, $\Theta(1)$ and $\Theta(n)$ are notational abuse and they represent arbitrary positive constant and arbitrary linear function respectively.

First, we guess the solution, $$ T(n)=O(n\log n) $$ which expanded, $$ T(n)\le cn\log n \quad \forall n\ge n_0,\text{where }c>0, n_0\in \mathbb N $$

Now, we want to prove $\forall nP(n)$, where $$ P(n):T(n)\le cn\log n \quad \forall n\ge n_0,\text{where }c>0, n_0\in \mathbb N $$

We will assume $P(k)$ for some $k$ and show $P(k+1)$. But what does it mean to assume $P(k)$ for some $k$ here? The expression $\forall k\ge n_0$ hurts by brain since we have taken $k$ to be some natural number and prefixing it with universal quantifier doesn't make sense to me.

How do we go about this?

Update

Actually the problem is with $P(n)$. It cannot contain quantifiers. It is atomic in First-order Logic.

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1 Answer 1

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What you asking about is mathematical induction theorem, more known as mathematical induction principle. It states, that $$\begin{cases}{} P(n_0) \text{ true} & \\ (\forall n \geqslant n_0)\big(P(n)\Rightarrow P(n+1)\big) & \end{cases} \Rightarrow (\forall n\geqslant n_0)P(n) $$

verbally it means, that for proving $(\forall n \geqslant n_0)P(n) $ is enough to prove two sentences: $P(n_0)$ and implication $(\forall n \geqslant n_0)\big(P(n)\Rightarrow P(n+1)\big) $. In last one $n$ is, so called, bound variable, dummy variable, and can be replaced with any other letter except $P$ and $n_0$. For example it same with $(\forall k \geqslant n_0)\big(P(k)\Rightarrow P(k+1)\big) $.

About update

Sentence under consideration is $T(n)=O(n\log n), n\to \infty\quad (2)$ and it's definition, of course contain quantifiers. As $n$ is bound variable in $(2)$, then it is not fully correct to denote it by $P(n)$, because, really, it does not contain $n$. Exact is to write $T\in O(f)$, where $f(x)=x \log x$.

Example of using:

Let me show how is it possible to use math. induction with sentences type $f\in O(g)\quad(3)$. Assume $f(n)=g(n)=n$. Formally $(3)$ is following: $$\exists c>0, \exists n_0 \in \mathbb{N}, \forall k > n_0, k\leqslant c\cdot k\quad (4)$$ All variables used are bound except $\mathbb{N}$. Let's choose $c=1=n_0$, denote by $P(k)$ sentence $ k\leqslant k$ and start proof of sentence $\forall k > 1,P(k)$, using math. induction with respect to $P$. Because holds $P(1)$ and $(\forall k \geqslant 1)\big(P(k)\Rightarrow P(k+1)\big)$, then proof is finished trivially.

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  • $\begingroup$ I understand what mathematical induction is. My confusion is in the following statement. "Assume $P(k)$ is true for some $k\in \mathbb N$". This is same as "Assume that $T(k)\le ck\log k \quad \forall k\ge n_0,$ where $c>0, n_0\in \mathbb N$ is true for some $k\in \mathbb N$". What does "$\forall k\ge n_0$ but for some $k\in \mathbb N$" mean? $\endgroup$
    – Jamāl
    Jul 20, 2022 at 12:47
  • $\begingroup$ As you pointed out, we want to prove $(\forall k \geqslant n_0)\big(P(k)\Rightarrow P(k+1)\big) $. For this, we assume $P(k)$ for some $k$ and not $\forall k\ge n_0 P(k)$. $\endgroup$
    – Jamāl
    Jul 20, 2022 at 12:52
  • $\begingroup$ Statement should be following "Consider any $k \geqslant n_0$. Assume $P(k)$ and prove $P(k+1)$". Word "some" in this context means "any", because you take "some", but without restriction other then $\geqslant n_0$. $\endgroup$
    – zkutch
    Jul 20, 2022 at 12:52
  • $\begingroup$ We do consider any $k\ge n_0$. But when we assume $P(k)$, $k$ is a fixed arbitrary natural number. Since after this we want to show $P(k+1)$, it wouldn't make any sense to let $k$ vary in the middle of proof. $\endgroup$
    – Jamāl
    Jul 20, 2022 at 12:54
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    $\begingroup$ Usually part with quantifiers are first in sentence: so, assume we want to prove $\exists c>0, \exists n_0 \in \mathbb{N}, \forall k > n_0, P(k)\quad (1)$. Now this can be proved with several methods. One of them induction. I speak about using induction for whole $(1)$. $\endgroup$
    – zkutch
    Jul 20, 2022 at 13:28

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