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\subsubsection*{Homework Solutions 1, Peter Anteater}

\begin{enumerate}
  \item $\displaystyle\int_0^2 x^2\,\dx =\left[\frac 13x^3\right]_0^2 =\frac 83$

  \item The truth table has the form
  \[
  	\begin{array}{c|c||c|c||c}
  		P&Q&\neg P&Q\vee\neg P&P\wedge (Q\vee \neg P)\\\hline
  		T&T&F&T&T\\
  		T&F&F&F&F\\
  		F&T&T&T&F\\
  		F&F&T&T&F
  	\end{array}
  \]
  Thus $P\wedge (Q\vee \neg P) \iff P\wedge Q$.\\
  
  Alternatively, this can be proved using the distributive laws:
  \begin{align*}
  	P\wedge (Q\vee \neg P) &\iff (P\wedge Q)\vee (P\wedge \neg P)\\
  	&\iff P\wedge Q
  \end{align*}
  since $P\wedge \neg P$ is a contradiction (always false).
\end{enumerate}

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