This site is being phased out.

# Intro to Higher Mathematics -- Spring 2016 -- final exam

Name:_________________________ $\qquad$ 12 problems, 170 points total

• Unless there is the word "prove" in the statement of the problem, all explanations are optional.

$\bullet$ 1. (10 points) Represent the following sentence as a logical expression:

If either $x$ or $y$ is negative, then so is $z$.

.

.

.

$\bullet$ 2. (10 points) Restate in plain English: $$\forall x \exists y \forall z (x=0\Rightarrow yz=0).$$

.

.

.

$\bullet$ 3. (10 points) Give the contrapositive of the following statement:

If either $x$ or $y$ is negative, then so is $z$.

.

.

.

$\bullet$ 4. (10 points) State the converse of the following: $$\forall x \exists y\in Y (\bar{A} \Rightarrow B ).$$

.

.

.

$\bullet$ 5. (10 points) State the negation of the following statement:

There are no real numbers $x$ and $y$ such that $x+y=1$ and $x-y=2$.

.

.

.

$\bullet$ 6. (20 points) Use induction to prove: $$1+2+3+...+n = \frac{n(n+1)}{2}.$$

.

.

.

.

.

.

$\bullet$ 7. (20 points) (a) Give examples, whenever possible, of the following functions, one onto and one one-to-one:

• $f:{\bf R}\to {\bf R}$;
• $f:{\bf R}\to {\bf Z}$;
• $f:{\bf Z}\to {\bf Z}_2$.

(b) List all bijections of the set $\{1,2,3\}$ to itself.

.

.

.

.

.

.

$\bullet$ 8. (20 points) Prove that the composition of two onto functions is onto.

.

.

.

.

.

.

$\bullet$ 9. (20 points) A group $G=\{a,b,c,d\}$ is given below by its table of operations. By renaming the elements of $G$ (like this: $a=3?,b=0?$, etc.) show that $G$ is the same as ${\bf Z}_4=\{0,1,2,3\}$: $$\begin{array}{c|cccc} +&a&b&c&d\\ \hline a&b&d&a&c\\ b&d&c&b&a\\ c&a&b&c&d\\ d&c&a&d&b \end{array}$$

.

.

.

.

.

.

$\bullet$ 10. (10 points) Functions $y=f(x)$ and $u=g(y)$ are given below by tables of some of their values. Present the composition $u=h(x)$ of these functions by a similar table: $$\begin{array}{c|c|c|c|c} x &0 &1 &2 &3 &4 \\ \hline y=f(x) &1 &1 &3 &0 &2 \end{array}$$ $$\begin{array}{c|c|c|c|c} y &0 &1 &2 &3 &4 \\ \hline u=g(y) &3 &1 &2 &1 &0 \end{array}$$

.

.

.

$\bullet$ 11. (10 points) Give an example of a metric on the set $X=\{A,B,C,D\}$.

.

.

.

$\bullet$ 12. (20 points) Prove the Triangle Inequality, i.e., $d(u,v)+d(v,w) \ge d(u,w)$, for the metric on the plane $X={\bf R}^2$ defined by:

• if $u\ne v$, then $d(u,v)=||u||+||v||$; and
• if $u=v$, then $d(u,v)=0$.

.

.

.

.

.

.