# Thread: Proving the Hypothetical Syllogism using Induction

1. ## Proving the Hypothetical Syllogism using Induction

Hi All,

First post here. thanks for the help in advance!

Okay first, the problem:

In Logic, the hypothetical syllogism says that given three statements, P, Q, and R, the hypothesis P $\rightarrow$ Q and Q $\rightarrow$ R logically imply P $\rightarrow$ R. Symbolically written ((P $\rightarrow$ Q) ^ (Q $\rightarrow$ R) $\Rightarrow$ (P $\rightarrow$ R))
Use induction to show that, given n statements ( $^A{1}$, $^A{2}$, .... $^A{n}$) that ( $^A{1}$ $\rightarrow$ $^A{2}$) ^ ( $^A{2}$ $\rightarrow$ $^A{3}$ ^ ... ^ ( $^A{n-1}$ $\rightarrow$ $^A{n}$) $\Rightarrow$ ( $^A{1}$ $\rightarrow$ $^A{n}$) for n $\geq$ 2

The problem I am having is that I am not sure how to work out an equation from this statement that I can start to use induction on. Also, I'm not sure if I should try to prove this using an inequality expression or an = expression. It would seem to me that an inequality would make more sense in an implication of this sort, but I am anything but positive on it.

I tried subbing in an arbitrary x for $^A{1}$, and assigning (x+1) to $^A{2}$, and using $^A{1}$ $\leq$ $^A{2}$, but my problem exploded and became unreasonable.

Can someone please help me find what recursive function or sum equation I should start with? I should be able to solve the proof from there with mathematical induction (also, do you suppose strong induction or standard will work in this case?)

Thanks again all!

-EDIT-

Here is what I have done so far. I think it proves this is true for $^A{n}$ when f(n)=n, but is this all I need to do? What about cases where f(n) is equal to something other than just n?

let $^A{n}$ = f(n) where:
f(1) = 1
f(n) = n for n $\geq$ 2

Step 1: (Base case) LHS $^A{2}$ = 2, and RHS f(2) = 2
Step 2: (Show $^A{n+1}$ = f(n+1)
$^A{n+1}$ = f(n+1)
$^A{n+1}$ = n+1 $\leftarrow$ induction step

so there is the proof for $^A{n+1}$, but something just feels wrong about this. have I solved this problem, or am I missing something here?