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!