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added 0921 notes and fixed hw3 errors

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Rushil Umaretiya 2023-09-21 15:15:21 -04:00
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@ -322,20 +322,15 @@ Prove that there are no solutions in integers \(x\) and \(y\) to the equation \(
|2x^2| &\leq 14\\
|5y^2| &\leq 14\\
\end{align*}
Our possible values for \(x\) are \(\{0,1,2,3\}\) and our possible values for \(y\) are \(\{0,1,2\}\).\\
Our possible values for \(x\) are \(\{0,1,2\}\) and our possible values for \(y\) are \(\{0,1\}\).\\
\begin{align*}
2(0)^2 + 5(0)^2 &= 0 + 0 = 0 &\neq 14&&(0,0)\\
2(0)^2 + 5(1)^2 &= 0 + 5 = 5 &\neq 14&&(0,1)\\
2(0)^2 + 5(2)^2 &= 0 + 20 = 20 &\neq 14&&(0,2)\\
2(1)^2 + 5(0)^2 &= 2 + 0 = 2 &\neq 14&&(1,0)\\
2(1)^2 + 5(1)^2 &= 2 + 5 = 7 &\neq 14&&(1,1)\\
2(1)^2 + 5(2)^2 &= 2 + 20 = 22 &\neq 14&&(1,2)\\
2(2)^2 + 5(0)^2 &= 8 + 0 = 8 &\neq 14&&(2,0)\\
2(2)^2 + 5(1)^2 &= 8 + 5 = 13 &\neq 14&&(2,1)\\
2(2)^2 + 5(2)^2 &= 8 + 20 = 28 &\neq 14&&(2,2)\\
2(3)^2 + 5(0)^2 &= 18 + 0 = 18 &\neq 14&&(3,0)\\
2(3)^2 + 5(1)^2 &= 18 + 5 = 23 &\neq 14&&(3,1)\\
2(3)^2 + 5(2)^2 &= 18 + 20 = 38 &\neq 14&&(3,2)\\
\end{align*}
Therefore, \(\forall x,y \in \AllIntegers(\neg P(x,y))\).
\end{proof}
@ -351,32 +346,37 @@ Prove that \(\sqrt[3]{2}\) is irrational.
\sqrt[3]{2} &= \frac{p}{q} && \text{Given } \neg P(x)\\
2 &= \frac{p^3}{q^3} && \text{Cubing both sides.}\\
2q^3 &= p^3 && \text{Multiplying both sides by } q^3.\\
\end{align*}
Let \(\bar{k} = q^3, \bar{k} \in \AllIntegers\).
\begin{align*}
2\bar{k} &= p^3 && \text{Algebraic Substitution.}\\
p^3 &\text{ is even.} && \text{Since } 2\bar{k} \text{ is even.}\\
p &\text{ is even.} && \text{Since } p^3 \text{ is even.}\\
\end{align*}
Given that p is even, Let \(\bar{k} \in \AllIntegers, p=2\bar{k}\).
\begin{align*}
2q^3&= (2\bar{k})^3 && \text{Algebraic Substitution.}\\
2q^3&= 8\bar{k}^3 && \text{Algebraic Equivalence.}\\
q^3&= 4\bar{k}^3 && \text{Algebraic Equivalence.}\\
\end{align*}
Since \(p\) is even, we have two cases where \(q\) is even and \(q\) is odd.\\
\begin{enumerate}
\item \(q\) is even.\\
\begin{align*}
2|p, 2|q&&\text{Since p and q are even.}\\
\gcd(p,q) \geq 2&&\text{Since p and q are even.}\\
\gcd(p,q) \geq 2&&\text{Since p and q are even.}
&&\bot\\
\end{align*}
We find the contradiction in the statement, since \(\gcd(p,q) \geq 2\) and \(p/q\) is in lowest terms.
\item \(q\) is odd.\\
\begin{align*}
\exists l \in \AllIntegers, p &= 2l + 1&&\text{Since p is odd.}\\
2q^3 &= (2l + 1)^3&&\text{Algebraic Substitution.}\\
2q^3 &= (4l^2 + 4l + 1)(2l + 1)&&\text{Algebraic Expansion.}\\
2q^3 &= 8l^3 + 12l^2 + 6l + 1&&\text{Algebraic Expansion.}\\
2q^3 &= 2(4l^3 + 6l^2 + 3l) + 1&&\text{Algebraic Equivalence.}\\
\exists l \in \AllIntegers, q &= 2l + 1&&\text{Since q is odd.}\\
(2l + 1)^3 &= 4\bar{k}^3&&\text{Algebraic Substitution.}\\
8l^3 + 12l^2 + 6l + 1 &= 4\bar{k}^3&&\text{Algebraic Expansion.}\\
8l^3 +12l^2 +6l-4\bar{k}^3 &= -1&&\text{Algebraic Equivalence.}\\
2(4l^3 + 6l^2 + 3l-2\bar{k}^3) &= -1&&\text{Algebraic Equivalence.}\\
\bar{l} \in \AllIntegers, \bar{l} &= 4l^3 + 6l^2 + 3l-2\bar{k}^3&&\text{Define } \bar{l}\\
2\bar{l} &= -1&&\text{Algebraic Equivalence.}
&&\bot\\
\end{align*}
We find the contradiction in the statement, since \(2q^3\) is even and \(2(4l^3 + 6l^2 + 3l) + 1\) is odd.
We find the contradiction in the statement, since \(2\bar{l}\) cannot equal \(-1\).
\end{enumerate}
Therefore, we have proved that \(\sqrt[3]{2}\) is irrational by contradiction.
\end{proof}

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\documentclass{article}
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\begin{document}
\maketitle
\pagebreak
\begin{proposition}
The sum of the first n, positive odd integers is the equation \(n^2\).
\end{proposition}
\begin{proof}
By induction
\begin{enumerate}
\item base case
\begin{align*}
P(1) &= 1^2 = 1\\
\end{align*}
\item inductive step
\begin{align*}
\text{Assume } P(n) = "1+3+5+...+2n-1 = n^2"\\
\text{WTS: } P(n+1) = "1+3+5+...+2n-1+2(n+1)-1 = (n+1)^2"\\
1+3+5+...+2n-1+2(n+1)-1 &= (n+1)^2\\
\end{align*}
\end{enumerate}
\end{proof}
\begin{proposition} All horses are the same color.\\
\emph{Note:} It suffices to show: P(n) = "All sets of n horses have the same color"\\
\begin{proof}
By induction
\begin{enumerate}
\item base case:\\
P(1) = "All sets of 1 horse have the same color"\\
\item inductive step: Assume P(n), i.e. every step of n horses have the same color\\
WTS: P(n+1), i.e. every set of n+1 horses have the same color\\
H = \(\{H_1, H_2, ..., H_n,H_{n+1}\}\) is a set of n+1 horses\\
\(H_1 = \{H_1, H_2, ..., H_n\}\) is a set of n horses\\
\(H_2 = \{H_2, H_3, ..., H_n,H_{n+1}\}\) is a set of n horses\\
\end{enumerate}
\end{proof}
\end{proposition}
\begin{theorem}
Given two sets, when \(n \neq 1\), when they both overlap and are disjoint, the union of the two sets is equal to the sum of the two sets.
\end{theorem}
\pagebreak
\section{Strong Induction}
This is what we were doing before:\\
\textbf{Weak Induction:}
\begin{enumerate}
\item base case
\item inductive step (\(P(n) \implies P(n+1)\))
\end{enumerate}
Trying to prove \(P(n) \forall n \in \AllNaturals\).\\\\
If \(P(n) \implies P(n+1)\) is too hard to show, instead try strong induction:
\begin{enumerate}
\item base case (Assume all steps before \(n+1\))
\item Assume \(P(k)\forall k \in \{1,2,...,n\}\) then try to show P(n+1)
\end{enumerate}
\begin{proposition}
A chocolate bar with n \(\geq\) 1 pieces can be broken into individual pieces by making n-1 breaks.
\end{proposition}
\begin{proof}
Using weak induction,
\begin{enumerate}
\item base case: n = 1\\
1 piece can be broken into individual pieces by making 0 breaks.
\item inductive step\\
Assume P(n)="a bar with n pieces can be broken into individual pieces by making n-1 breaks"\\
WTS: P(n+1)="a bar with n+1 pieces can be broken into individual pieces by making n breaks"\\
The issue is that we need to know that everything from P(n) to P(1) works.\\
\textbf{Since we cannot prove this for an arbitrary n, we must use strong induction.}\\
\end{enumerate}
\end{proof}
\begin{proof}
Using strong induction,
\begin{enumerate}
\item base case: n = 1\\
1 piece can be broken into individual pieces by making 0 breaks.
\item inductive step\\
Assume P(k) \(\forall k \in \{1,2,...,n\}\)\\
WTS: P(n+1)="a bar with n+1 pieces can be broken into individual pieces by making n breaks"\\
\begin{enumerate}
\item Consider an arbitrary bar of n+1 size. Break the bar into two pieces
\begin{enumerate}
\item One piece has k pieces
\item The other piece has \((n+1)-k\) pieces
\end{enumerate}
\item Assuming \(P(k)\), the first piece can be broken into individual pieces by making \(k-1\) breaks.
\item \(P(n+1-k)\) will require \(n+1-k-1=n-k\) breaks.
\end{enumerate}
\(\therefore\) The total number of breaks is \(1+(k-1)+(n-k)=n\).
\end{enumerate}
\end{proof}
\pagebreak
\begin{theorem}
It is true that strong induction \(\rightarrow\) weak induction
\end{theorem}
\begin{theorem}
Fundamental Theorem of Arithmetic:
\begin{align*}
\forall n \in \AllNaturals - {0,1}, \text{n is either prime or can be written as a product of primes.}\\
\end{align*}
\end{theorem}
\begin{proof}
Using strong induction,
\begin{enumerate}
\item base case: n = 2\\
2 is prime.
\item inductive step: Assume P(k) \(\forall k \in \{2,3,...,n\}\)\\
WTS: P(n+1)\\
We can prove this by cases:
\begin{enumerate}
\item n+1 is prime:
It can be expressed as \(1 \times (n+1)\)
\item n+1 is not prime\\
\begin{align*}
\exists l,w \in \AllIntegers, n+1 = lw\\
1 < l, w < n+1\\
\end{align*}
So P(l) = T, P(w) = T, therefore:\\
\(l = p_1^{k_1}\times p_2^{k_2} \times ... \times p_n^{k_n}\), where \(p_1, p_2, ..., p_n\) are primes and \(k\in \AllNaturals\).\\
\(w = q_1^{j_1}\times q_2^{j_2} \times ... \times q_m^{j_m}\), where \(q_1, q_2, ..., q_m\) are primes and \(j\in \AllNaturals\).\\\\
Given \(l\) and \(w\), we can find \(n+1\) by multiplying them together.\\
\begin{align*}
n+1 &= lw\\
n+1 &= (p_1^{k_1}\times p_2^{k_2} \times ... \times p_n^{k_n}) (q_1^{j_1}\times q_2^{j_2} \times ... \times q_m^{j_m})\\
\end{align*}
\end{enumerate}
\end{enumerate}
\(n+1\) is a product of primes.\\
\end{proof}
\pagebreak
\begin{proposition}
Consider the sequence \(a_1=0, a_2=1, a_n=2a_{n-1}-a_{n-2}\). Prove \(a_n = n-1\).
\end{proposition}
\begin{proof}
Using strong induction,
\begin{enumerate}
\item bsae case: n = 1, n=2\\
\(n=1: a_1 = 0 = 1-1\)\\
\end{enumerate}
\end{proof}
\end{document}