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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:

It is possible for both $x$ and $y$ to be positive.

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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:

If either $x$ or $y$ is $0$, then so is $xy$.

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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:

If either $x$ or $y$ is $0$, then so is $xy$.

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

All sets are disjoint.

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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: $$(1 + x)^n \ge 1 + nx.$$