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# 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:

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

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

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

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

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

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$\bullet$ **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.

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

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$\bullet$ **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 ${\bf Z}_4=\{0,1,2,3\}$:
$$\begin{array}{c|cccc}
+&a&b&c&d\\
\hline
a&b&d&a&c\\
b&d&c&b&a\\
c&a&b&c&d\\
d&c&a&d&b
\end{array}$$

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$\bullet$ **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:
$$\begin{array}{c|c|c|c|c}
x &0 &1 &2 &3 &4 \\
\hline
y=f(x) &1 &1 &3 &0 &2 \end{array}$$
$$\begin{array}{c|c|c|c|c}
y &0 &1 &2 &3 &4 \\
\hline
u=g(y) &3 &1 &2 &1 &0 \end{array}$$

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

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$\bullet$ **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 $d(u,v)=0$.

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