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:

It is impossible for both $x$ and $y$ to be negative.

$\bullet$ 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:

All students in this class will get an A.

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

In a class of $10$, there are at least $2$ students with the same major.

$\bullet$ 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:

There are real numbers $x$ and $y$, both positive, such that $x+y=1$ and $x-y=2$.

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