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 Q and Q R logically imply P R. Symbolically written ((P Q) ^ (Q R) (P R))
Use induction to show that, given n statements ( , , .... ) that ( ) ^ ( ^ ... ^ ( ) ( ) for n 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 , and assigning (x+1) to , and using , 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 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 = f(n) where:
f(1) = 1
f(n) = n for n 2
Step 1: (Base case) LHS = 2, and RHS f(2) = 2
Step 2: (Show = f(n+1)
= f(n+1)
= n+1 induction step
so there is the proof for , but something just feels wrong about this. have I solved this problem, or am I missing something here?