Abstract Algebra IBL Wiki

Automorphisms Of A Directed Square

(:note:) Please work on the next problem again. I would like you to come to class with a written proof that uses induction. Here's a different problem, but the proof technique is related.

!!!!!Example Provide a proof of the following claim. *'Claim': For every positive integer , the sum of differentiable functions is differentiable.

'Proof': We proceed by induction. We know that if is differentiable, then the sum is differentiable (this is the base case). If you are not happy with this base case, then we also know that if and are differentiable functions, then is a differentiable function (something we proved in first semester calculus). So we have shown that the claim is true if or .

We now assume that the statement is true when . So we know that if we have differentiable functions, then their sum is differentiable. We need to show that if are differentiable functions, then their sum is differentiable. If we let for each between 1 and , and then we combine the last two functions to give use , then we have functions that are all differentiable. The last function is differentiable because it is the sum of two differentiable functions. So we now have $skk+1$ differentiable functions is differentiable.

We've now shown that if the claim is true for , then the claim is true for . By the principle of mathematical induction, our claim is true for each positive integer .

(:noteend:)

The Composition Of Permutations Is A Permutation

The Composition Of Permutations Is A Permutation

Simple Matrix Encryption

Composition Combination Of Permutations

Simple Shift Repetition Game

The Span Of A Simple Shift

!!!! Reading Assignment and First Write Up Start by reading The elements of Style for Proofs. For Monday, please pick one of the problems that we have discussed in class on either Wed or Fri, and write a solution. Use complete sentences and make sure you justify your work. Make sure that you use your own words to give your solutions. The goal of this portion of the course is to help you improve your own personal writing. When you have completed your problem and you are ready to submit it, please type [@!Submit@] (yes it has an exclamation point) at the bottom of the page. This will let me know that it is ready for grading.