Abstract Algebra IBL Wiki

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[[Problem:Automorphisms Of A Directed Square]]
{{page>Problem:Automorphisms Of A Directed Square#problem}}

(: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 $n$, the sum of $n$ differentiable functions is differentiable. 

'''Proof''': We proceed by induction.  We know that if $f_1$ is differentiable, then the sum $f_1$ is differentiable (this is the base case).  If you are not happy with this base case, then we also know that if $f_1$ and $f_2$ are differentiable functions, then $f_1+f_2$ is a differentiable function (something we proved in first semester calculus).  So we have shown that the claim is true if $n=1$ or $n=2$. 

We now assume that the statement is true when $n=k$. So we know that if we have $n$ differentiable functions, then their sum is differentiable.  We need to show that if $g_1, g_2, \ldots g_k,g_{k+1}$ are $k+1$ differentiable functions, then their sum $$s=g_1+g_2+\cdots+g_{n-1}+g_n+g_{n-1}$$ is differentiable.  If we let $h_i=g_i$ for each $i$ between 1 and $k-1$, and then we combine the last two functions to give use $h_k=g_k+g_{k+1}$, then we have $k$ functions $h_1, h_2, \ldots h_k$ that are all differentiable. The last function $h_k$ is differentiable because it is the sum of two differentiable functions. So we now have 
$$
\begin{align*}
s&=g_1+g_2+\cdots+g_{k-1}+(g_k+g_{k-1})\\
&h_1+h_2+\cdots+h_{k-1}+(h_k).
\end{align*}
$$
Since $s$ is the sum of $k$ differentiable functions, our induction assumption tells us it is differentiable.  This shows that the sum of $k+1$ differentiable functions is  differentiable.  

We've now shown that if the claim is true for $n=k$, then the claim is true for $n=k+1$.  By the principle of mathematical induction, our claim is true for each positive integer $n$. 

(:noteend:)

[[Theorem:The Composition Of Permutations Is A Permutation]]
{{page>Theorem:The Composition Of Permutations Is A Permutation#theorem}}


[[Problem:The Composition Of Permutations Is A Permutation]]
{{page>Problem:The Composition Of Permutations Is A Permutation#problem}}


[[Problem:Simple Matrix Encryption]]
{{page>Problem:Simple Matrix Encryption#problem}}


[[Definition:Composition Combination Of Permutations]]
{{page>Definition:Composition Combination Of Permutations#definition}}


[[Problem:Simple Shift Repetition Game]]
{{page>Problem:Simple Shift Repetition Game#problem}}


[[Problem:The Span Of A Simple Shift]]
{{page>Problem:The Span Of A Simple Shift#problem}}



!!!! Reading Assignment and First Write Up
Start by reading [[http://bmw.byuimath.com/aa/doku.php?id=elements_of_style_for_proofs|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.



{{tag>tarafife ben}}
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