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\noindent{\Large{\textsc{Intro to Abstract Mathematics\hfill Assignment 3.2  \\ Spring 2020 \hfill Due February 12}}}

\large

\vspace{1cm}


\begin{exercise}
Let $a, b \in \N$.  The {\em greatest common divisor} of $a$ and $b$, denoted $\gcd(a,b)$, is defined to be the greatest element in the set $\{ d \in \N \, | \, d \text{ divides both } a \text{ and } b \}$.  Prove that $\gcd(a,b) = \gcd(a+b,a)$.  [{\em Suggestion:}  Show that the sets defining the two gcds are the same.]


\end{exercise}

\begin{exercise}
The {\em symmetric difference} of two sets $A$ and $B$ is defined to be
\[
A \Delta B = (A \setminus B) \cup (B \setminus A).
\]
Verify the following properties of the symmetric difference.
\begin{itemize}

\item[\bf a.] $A \Delta B = B \Delta A$. 

\item[\bf b.]  $A \Delta B = (A \cup B) \setminus (A \cap B)$.

\item[\bf c.]  $A \Delta B = A \cup B$ if and only if $A$ and $B$ are disjoint.

\end{itemize}

\end{exercise}

\begin{exercise}
Let $A, B, C$ be sets.
\begin{itemize}

\item[\bf a.]  Show that
\[
(A \Delta B) \Delta C = \left[(A \cup B \cup C) \setminus ((A \cap B) \cup (A \cap C) \cup (B \cap C)) \right] \cup (A \cap B \cap C).
\]

\item[\bf b.]  Conclude that $(A \Delta B) \Delta C = (B \Delta C) \Delta A$.

\item[\bf c.]  Use {\bf 2a} and part {\bf b} to show that $(A \Delta B)\Delta C = A \Delta (B \Delta C)$.

\end{itemize}
\end{exercise}


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