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Extra Credit &\huge Proof Practice \small take 2 &\normalsize TCSS 321 \\
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\noindent Prove or disprove the following:
\begin{enumerate}
%5.1 #4
\item For all positive integers $n$, $\displaystyle{\sum_{i=1}^n(i^3)=\left(\frac{n(n+1)}{2}\right)^2}$

%1.8 #6
\item The number $5x+5y$ is an odd integer when $x$ and $y$ are integers of opposite parity.

%2.1 #46
\item Let $S$ be the set that contains a set $x$ if the set $x$ does not belong to itself, so $S=\{x \vert x\not\in x\}$. $S$ is a member of $S$.

%2.1 #46
\item Let $S$ be the set that contains a set $x$ if the set $x$ does not belong to itself, so $S=\{x \vert x\not\in x\}$. $S$ is a not a member of $S$.

%5.1 #31
\item 2 divides $n^2+n$ whenever $n$ is a positive integer.

%2.2
\item If $A$, $B$, and $C$ are sets, then $\overline{A} \cap \overline{B}\cap \overline{C}=\emptyset$.

%5.1 #22
\item If $n$ is an integer greater than 4, then $2^n>n^2$.

%1.7 \#26
\item For a positive integer $n$, $n$ is even if and only if $7n+4$ is even.

%5.2 \#10
\item Consider a rectangular checkerboard that you'd like to divide into its constituent squares.  You can either make (complete) cuts along the vertical lines or horizontal lines separating the squares.  There is no formula to determine how many cuts you must make to break the checkerboard into the $n$ separate squares.

%2.2 \#46
\item If $A$, $B$, and $C$ are sets then:\\ $\vert A \cup B \cup C\vert=\vert A\vert +\vert B\vert+\vert C\vert -\vert A\cap B \vert -\vert A\cap C \vert -\vert B\cap C\vert +\vert A\cap B\cap C\vert$.

%5.1 #12
\item For all nonnegative integers $n$, $\displaystyle{\sum_{i=0}^n\left(\frac{-1}{2}\right)^i=\frac{2^{n+1}+(-1)^n}{3\cdot 2^n}}$.


%5.1 \#20
\item If $n$ is an integer greater than 6, then $3^n<n!$.



\end{enumerate}
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