EQ 8 Proof By Induction

(1) Axiom: Mathematical Induction Over $\mathcal{N}$. Form 1

( $\forall$ n:$\mathcal{N}$ |: ( $\forall$i | 0 $\leq$ i $\lt$ n: $P_i$) $\Rightarrow$ $P_n$ ) $\Rightarrow$ ( $\forall$ n : $\mathcal{N}$ : $P_n$)

used to prove Universal Quantification By Induction


(2) Theorem: Mathematical Induction Over $\mathcal{N}$. Form 2

( $\forall$ n:$\mathcal{N}$ |: ( $\forall$i | 0 $\leq$ i $\lt$ n: $P_i$) $\Rightarrow$ $P_n$ ) $\equiv$ ( $\forall$ n : $\mathcal{N}$ : $P_n$)

used to prove properties of Induction

(3) Theorem: Mathematical Induction Over $\mathcal{N}$. Form 3

$P_0$ $\wedge$ ( $\forall$ n:$\mathcal{N}$ |: ( $\forall$i | 0 $\leq$ i $\leq$ n: $P_i$ ) $\Rightarrow$ $P_{n+1}$ ) $\Rightarrow$ ( $\forall$ n : $\mathcal{N}$ : $P_n$)

restatement of (1) used for inductive proofs

(4) Parts of Mathematical Induction axiom

$P_0$ is the base case

( $\forall$ n:$\mathcal{N}$ |: ( $\forall$i | 0 $\leq$ i $\leq$ n: $P_i$ ) $\Rightarrow$ $P_{n+1}$ ) $\Rightarrow$ ( $\forall$ n : $\mathcal{N}$ : $P_n$) is the inductive case

( $\forall$ n:$\mathcal{N}$ |: ( $\forall$i | 0 $\leq$ i $\leq$ n: $P_i$ ) $\Rightarrow$ $P_{n+1}$ ) is the inductive hypothesis

(5) Normal Proof Method

(1) Prove Base Case. $P_0$

(2) Assume arbitrary n $\ge$ 0

(3) Assume Inductive Case ( $\forall$i | 0 $\leq$ i $\leq$ n: $P_i$ )

(4) Prove $P_{n+1}$


(6) Induction starting at other integers

Let a sequence of integers start at $n_0$ Then Inductive Theorem is

$P_{n_0}$ $\wedge$ ( $\forall$ n:$n_0 \leq n$|: ( $\forall$i | $n_0 \leq$ i $\leq$ n: $P_i$ ) $\Rightarrow$ $P_{n+1}$ ) $\Rightarrow$ ( $\forall$ n : $n_0 \leq n$ : $P_n$)