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# Intro to Higher Mathematics -- Spring 2018 -- midterm

**MATH 300 -- Spring 2018 -- midterm exam**

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

- Except for the last problem, all explanations are optional but encouraged.

$\bullet$ **1.** Provide the English sentence represented by this logical expression with omitted parentheses:
$$ P \wedge Q \vee \bar{R},$$
where

- $P=$"I will buy the pants",
- $Q=$"I will buy the shirt",
- $R=$"I will buy the shoes".

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$\bullet$ **2.** Represent the following sentence as a logical expression:

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$\bullet$ **3.** Restate the following in terms of inclusions of sets:
$$\forall x \bigg( x\in X \text{ or } x\not\in Y \Longleftrightarrow x\in A \text{ and } x\not\in B \bigg).$$

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$\bullet$ **4.** Give the contrapositive of the following statement:

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$\bullet$ **5.** Give a counter-example, if possible, for the following statement:
$$\forall x \exists y \big( x+y=1 \big).$$

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$\bullet$ **6.** State the converse of the following statement:

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$\bullet$ **7.** State the negation of the following statement:

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$\bullet$ **8.** Functions $y=f(x)$ and $u=g(y)$ are given below by tables of their values. Find the inverse of their composition $h=g\circ f$, if possible:
$$\begin{array}{c|c|c|c|c}
x &0 &1 &2 &3 &4 \\
\hline
y=f(x) &1 &1 &3 &0 &2 \end{array}$$
$$\begin{array}{c|c|c|c|c}
y &0 &1 &2 &3 &2 \\
\hline
z=g(y) &3 &1 &2 &1 &0 \end{array}$$

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$\bullet$ **9.** Suppose a set $X$ has $2$ elements and a set $Y$ has $3$ elements. Consider the functions $f:X\to Y$ and list separately: (a) all onto functions, (b) all one-to-one functions, (c) all bijections.

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$\bullet$ **10.** Use induction to prove that for any positive integer $n$ and any $x > −1$, we have:
$$(1 + x)^n \ge 1 + nx.$$