proving associative property of any two vectors

Problem:Prove that any two ways of associating a sum of any number of vectors give the same sum. (Hint. Use induction on the number of vectors.)

Proof: Base Case: Let there be three vectors namely **v**_{1},v_{2},v_{3}. We see that **(v**_{1+}v_{2})_{+}v_{3=}**v**_{1+}(v_{2}_{+}v_{3}).

Inductive Case: Let k be the number of vectors where k∈N. We want to assume that for **v**_{1...}v_{k}that we could associate them any two ways and get the same sum. We want to show for k+1 vectors that the same thing holds.

Case 1: Odd number of k+1 vectors

**(** **(v**_{1+}v_{2})+...+(v_{k-1}+v_{k}))+v_{k+1 }=(**v**_{1+} (**v**_{2}_{+}v_{3})+...+(**v**_{k}+V_{k+1}))=v1+** ((****v**_{2}_{+}v_{3})+...+(**v**_{k}+V_{k+1}))

Case 2: Even Number of k+1 vectors

**((****v**_{1+}v_{2})+...+(v_{k-2}+v_{k-1})+v_{k})+v_{k+1}=(v_{1}+(v_{2}+v_{3})+..+(v_{k-1}+v_{k}))+v_{k+1}=(v_{1}+v_{2})+...+(v_{k}+v_{k+1})=v_{1}+((v_{2}+v_{3})+..+(v_{k}+v_{k+1}))

Hence by induction it holds for k+1 vectors.

Did I do it correctly?

Re: proving associative property of any two vectors

you've only shown 3 out of the many, many cases for k odd, and 4 cases for k even. this particular proof is a pain, because the number of cases grows rapidly as k does.

let's say S is a sum (associated arbitrarily) of k+1 vectors. we want to show that S = ((...(v_{1}+v_{2})+v_{3})+...+v_{k})+v_{k+1}. we will assume this is true for any sum of j vectors with: 3 ≤ j ≤ k.

we can write:

S = A+B, for two sums A and B, and the largest either A or B can be is a sum of k vectors.

applying our induction hypothesis to B we have:

B = ((...(v_{j}+v_{j+1})+v_{j+2})+...+v_{k})+v_{k+1}, for some v_{j}.

applying our induction hypothesis to A, we have:

A = ((...(v_{1}+v_{2})+v_{3})+...+v_{j-2})+v_{j-1}.

if we let C = ((...(v_{j}+v_{j+1})+...+v_{k-2})+v_{k},

we have A+B = A+(C+v_{k+1}) = (A+C)+v_{k+1}.

applying the induction hypothesis to A+C, we have:

A+C = ((...(v_{1}+v_{2})+v_{3}+...+v_{k-1})+v_{k},

hence S = (A+C)+v_{k+1} = ((...(v_{1}+v_{2})+v_{3})+...+v_{k})+v_{k+1}, as desired.

(the problem with your proof is that association can occur with more than "adjacent pairs", you can have sums like:

((v_{1}+v_{2})+v_{3})+(v_{4}+v_{5}) or

(v_{1}+(v_{2}+v_{3})+v_{4})+v_{5} and so on).

Re: proving associative property of any two vectors

oh ok thanks. this makes alot of sense now. i didn't even think of that.