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\begin{document}
\begin{center}
\Large{Exercises for Module 1: Proofs}
\end{center}


1.  Prove De Morgan's Laws for propositions: $\neg (P \wedge Q) = \neg P \vee \neg Q$ and $\neg (P \vee Q) = \neg P \wedge \neg Q$ (Hint: use truth tables).

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2. Write the following statements and their negations using logical quantifier notation and then prove or disprove them: \\
(i) Every odd integer is divisible by three.

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(ii) For any real number, twice its square plus twice itself plus six is greater than or equal to five. \textit{(You may assume knowledge of calculus.)}

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(iii) Every integer can be written as a unique difference of two natural numbers.


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3. Prove the following statements: \\
(i) If $a | b$ and $a,b \in \N$ (positive integers), then $a \leq b$.

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(ii) If $a | b$ and $a | c$, then $a | (x b + y c)$, where $a,b,c,x,y \in \Z$.

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(iii) Let $a,b,n \in \Z$. If $n$ does not divide the product $ab$, then $n$ does not divide $a$ and $n$ does not divide $b$.

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4. Prove that for all integers $n \geq 1$, $3|(2^{2n}-1)$.

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5. Prove the Fundamental Theorem of Arithmetic, that every integer $n \geq 2$ has a unique prime factorization (i.e. prove that the prime factorization from the last proof is unique).

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\end{document}


