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# Midterm

**MATH300 -- Spring 2016 -- midterm**

Name:_________________________ $\qquad$ 9 problems, 100 points total

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

$\bullet$ **1.** Provide the English sentence represented by this logical expression:
$$\neg ( P \wedge \bar{Q}),$$
where

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

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

$\bullet$ **3.** Restate the following in terms of inclusion of sets:
$$x\in X \text{ or } y\not\in Y \Leftrightarrow a\not\in A \text{ and } b\in B.$$

$\bullet$ **4.** Restate in plane English:
$$\forall x \exists y \exists z (x>0\Rightarrow yz<0).$$

$\bullet$ **5.** Give the contrapositive of the following statement:

$\bullet$ **6.** State the hypothesis and the conclusion of the following:

$\bullet$ **7.** State the converse of the following:
$$\forall x \exists y\in Y (A\Rightarrow \bar{B} ).$$

$\bullet$ **8.** State the negation of the following statement:

$\bullet$ **9.** (20 points) Use induction to prove:
$$2^0+2^1+...+2^n=2^{n+1}-1.$$