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# Intro to Higher Mathematics -- Spring 2018 -- final exam

**MATH 300 -- Spring 2018 -- final exam**

Name:_________________________ $\qquad$ 7 problems, 10 points each

- Write the problems in the given order and start each problem at the top of the page.
- Show enough work to justify your answers.

$\bullet$ **1.** State the following: (a) the Pythagorean Theorem, (b) its contrapositive, (c) its converse, (d) its negation.

$\bullet$ **2.** Below is the addition (multiplication) table of a group. Complete it.
$$\begin{array}{l|ll}
&a&b&c&d\\
\hline
a&d&c\\
b&&&a\\
c\\
d
\end{array}$$

$\bullet$ **3.** Recall that a topology on a finite set $X$ is a collection of subsets of $X$ that contains $X$ and $\emptyset$ and is closed under unions and intersections. List all possible topologies on the set of three elements, $X=\{1,2,3\}$. Ignore permutations.

$\bullet$ **4.** Give an example of a function $f:{\bf Z}\to {\bf Z}$ for each of the following: (a) $f$ is onto but not one-to-one, (b) $f$ is one-to-one but not onto, (c) $f$ is both onto and one-to-one, (d) $f$ is neither one-to-one nor onto.

$\bullet$ **5.** Use induction to prove that for any positive integer $n$, we have:
$$1+3+5+...+(2n-1)=n^2.$$

$\bullet$ **6.** Prove that the composition of two onto functions is onto.

$\bullet$ **7.** Find a flaw in the following proof. *Statement:* For any $x,y,z\in {\bf Z}$ such that $3x+5y=7z$, if at least one of $x,y,z$ is odd then at least one of them is even. *Proof:* Suppose $x,y,z$ satisfy $3x+5y=7z$. Assume, to the contrary, that none of $x,y,z$ is odd and none of $x,y,z$ is even. This is impossible!