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\centerline{MATH 243 -- Mathematical Structures}
\centerline{Quiz 3 -- September 22, 2017}
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Your Name: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
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{\it Directions}
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Do all work and answer all questions on this sheet.  There are 30 possible points, 
distributed as indicated.    
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\begin{enumerate}
\item[A)]  (15)  State the three Peano Axioms for $\mathbb{N}$.  
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\item[B)]  (15)  Prove by induction:  For all natural numbers $n \ge 1$, 
$1 + 3 + \cdots + (2n - 1) = n^2$.  (Note:  the ``base case'' will be $n = 1$ here, although
you could also make the formula work for $n = 0$ with a convention about what the sum means in that case.)
Prove your assertion.    
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