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Problem Set 2

Problem Set 2
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Problem Set 2

 

 

  1. Analyze the logical forms of the following statements.

     

    1. If anyone in the dorm has the measles, then everyone who has a friend in the dorm will have to be quarantined.

       

    2. Jane saw a bear, and Roger saw one too.

       

    3. Every number that is larger than $ x$ is larger than $ y.$

       

    4. If there is a number $ x$ such that $ x^{2}+5x=w$ and there is a number $ y$ such that $ 4-y^{2}=w,$ then $ w$ is between $ -10$ and $ 10.$

       

    5. Anyone who has bought a Rolls Royce with cash must have a rich uncle.

       

    6. If nobody failed then test, then everybod who got an A will tutor someone who got a D.

     

  2. Consider the following incorrect theorem:

     

     

    Theorem 1   Suppose $ n$ is a natural number larger than $ 2$ and $ n$ is not a prime number. Then $ 2n+13$ is not a prime number.

     

     

    1. What are the hyptheses and conclusion of this theorem?

       

    2. Show that the theorem is incorrect by finding a counterexample.

     

  3. Suppose $ A\backslash B\subseteq C\cap D$ and $ x\in A.$ Prove that if $ % x\notin D$ then $ x\in B.$

     

  4. Suppose $ x$ is a real number and $ x\neq 0.$ Prove that if $ \frac{\sqrt [3]{x}+5}{x^{2}+6}=\frac{1}{x}$ then $ x\neq 8.$

     

  5. Suppose $ a,b,c,$ and $ d$ are real numbers, $ 0<a<b$ and $ d>0.$ Prove that if $ ac\geq bd$ then $ c>d.$

     

  6. Suppose $ x$ and $ y$ are real numbers and $ 3x+2y\leq 5.$ Prove that if $ x>1$ then $ y<1.$

     

  7. Consider the following theorem.

     

     

    Theorem 2   Suppose $ x$ is a real number and $ x\neq 4.$ If $ \frac{2x-5}{x-4}=3$ then $ % x=7.$

     

     

    1. What wrong with the following proof of the theorem?

       

       

      Proof. Suppose $ x=7.$ Then $ \frac{2x-5}{x-4}=\frac{2\left( 7\right) -5}{7-4}=\frac{9}{3}=3.$ Therefore if $ \frac{2x-5}{x-4}=3$ then $ x=7.$ $ \qedsymbol$

       

       

    2. Give a correct proof of the theorem.

     

  8. Suppose $ A\subseteq C$ and $ B$ and $ C$ are disjoint. Prove that if $ % x\in A$ then $ x\notin B.$

     

  9. Use proof by contradiction to prove the following theorem.

     

     

    Theorem 3   Suppose $ A\cap C\subseteq B$ and $ a\in C.$ Prove that $ a\notin A\backslash B. $

     

     

  10. Use proof by contradiction to prove the following theorem.

     

     

    Theorem 4   Suppose $ A\subseteq B,a\in A,$ and $ a$ and $ b$ are not both elements of $ B.$ Prove that $ b\notin B.$

     

     

  11. Consider the following incorrect theorem.

     

     

    Theorem 5   Suppose $ x$ and $ y$ are real numbers and $ x+y=10.$ Then $ x\neq 3$ and $ y\neq 8.$

     

     

    1. What is wrong with the following proof of the theorem?

       

       

      Proof. Supose the conclusion of the theorem is false. Then $ x=3$ and $ y=8.$ But then $ x+y=11$ , which contradicts the given information that $ x+y=10.$ Therefore, the conclusion must be true. $ \qedsymbol$

       

       

    2. Show that the theorem is incorrect by finding a counterexample.

     

  12. Prove that if $ A$ and $ B\backslash C$ are disjoint, then $ A\cap B\subseteq C$

     

  13. Suppose $ x$ is a real number.

     

    1. Prove that if $ x\neq 1$ then there is a real number $ y$ such that $ % \frac{y+1}{y-2}=x$

       

    2. Prove that if there is a real number $ y$ such that $ \frac{y+1}{y-2}=x$ then $ x\neq 1.$

     

  14. Prove that for every real number $ x$ , if $ x>2$ then there is a real number $ y$ such that $ y+\frac{1}{y}=x$

     

  15. Prove that if $ A\subseteq B$ and $ A\subseteq C$ then $ A\subseteq B\cap C$

     

  16. Suppose $ A\subseteq B$ . Prove that for every set $ C,$ $ C\backslash B\subseteq C\backslash A.$

     

  17. Prove

     

    1. A sufficient condition for the demand for a good to incresae when its price falls is that it is a normal good.

       

    2. A necessary but not sufficient condition for the demand for a good to decrease when its price falls is that it is an inferior good.

 

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Jennifer Anne Thacher 2008-09-05
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