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Intro to Higher Mathematics -- Spring 2016 -- midterm

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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: $$\forall x \bigg( x\in X \text{ or } x\not\in Y \Leftrightarrow x\in A \text{ and } x\not\in B \bigg).$$

$\bullet$ 4. Restate in plain 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: $$2^0+2^1+...+2^n=2^{n+1}-1.$$