Skip to content. | Skip to navigation

Personal tools
Log in
Sections
You are here: Home › Teaching › Grad. Math Econ › Problem Sets › PS1

Problem Set 1

next up previous
Next: About this document ...

Problem Set 1

  1. Let $ S$ stand for the statment "Steve is happy" and $ G$ for "George is happy". Which English sentences are represented by the following expressions?

    1. $ (S \vee G) \wedge (\lnot S \vee \lnot G)$
    2. $ [S \vee (G \wedge \lnot S)] \vee \lnot G$
    3. $ S \vee [G \wedge (\lnot S \vee \lnot G)]$

  2. Translate the following statements into symbolic notation.

    1. We'll have either a reading assignment or homework problems, but we won't have both homework problems and a test.

    2. Either John and Bill are both telling the truth or neither of them is.

  3. Make truth tables for the following formulas

    1. $ \lnot [P \wedge (Q \vee \lnot P)]$
    2. $ (P \vee Q) \wedge (\lnot P \vee R)$

  4. Find a formula using only the connectives $ \wedge $ and $ \lnot $ that is equivalent to $ P\vee Q.$ Justify your answer with a truth table.

  5. Use a truth table to determine whether the following statement is valid. The butler and the cook are not both innocent. Either the butler is lying or the cook is innocent. Therefore, the butler is either lying or guilty.

  6. Use truth tables to determine which of the following statements are equivalent to each other.

    1. $ \left( P\wedge Q\right) \vee \left( \lnot P\wedge \lnot Q\right) $

    2. $ \left( P\vee \lnot Q\right) \wedge \left( Q\wedge \lnot P\right) $

  7. Find simpler formulas equivalent to these formulas.

    1. $ P\vee \left( Q\wedge \lnot P\right) $

    2. $ \lnot \left( \lnot P\wedge \lnot Q\right) $

  8. Are these statements tautologies, contradictions, or neither?

    1. $ P\vee \left( Q\vee \lnot P\right) $

    2. $ P\wedge \lnot \left( Q\vee \lnot Q\right) $

  9. Let $ A=\left\{ 1,3,12,35\right\} ,B=\left\{ 3,7,12,20\right\} ,$ and $ % C=\left\{ x\vert\text{ }x\text{ is a prime number}\right\} .$ List the elements of the following sets. Are any of the sets below disjoint from any of the others? Are any of the sets below subsets of any others?

    1. $ A\cap B$

    2. $ \left( A\cup B\right) \backslash C$

  10. Use Venn diagrams to verify the following identities:

    1. $ (A \cup B) \backslash C = (A \backslash C) \cup (B \backslash C)$
    2. $ A \cup (B \backslash C) = (A \cup B) \backslash (C \backslash A)$

  11. Are either of the following statements equivalent?

    1. If it's raining then the game has been canceled and if it's snowing then the game has been canceled.

    2. If it's neither raining nor snowing then the game hasn't been canceled.

  12. Analyze the logical forms of the following statements

    1. If this gas either has an unpleasant smell or is not explosive, then it isn't hydrogen.

    2. Mary will sell her house only if she can get a good price and find a nice apartment.

    3. $ x$ and $ y$ are men, and either $ x$ is taller than $ y$ or $ y$ is taller than $ x$ .

    4. Either $ x$ or $ y$ has brown eyes, and either $ x$ or $ y$ has red hair.

  13. Analyze the following statements and determine whether each is equivalent or the converse of: ''If it is raining then it is windy and the sun is not shining''.

    1. It is windy and not sunny only if it is raining

    2. Rain is a necessary condition for wind with no sunshine.

  14. Use truth tables to determine whether or not the following statements are valid:

    1. Either sales or expenses will go up. If sales go up, then the boss will be happy. If expenses go up then the boss will be unhappy. Therefore, sales and expenses will not both go up.

    2. If the tax rate and the unemployment rate both go up, then there will be a recession. If the GNP goes up, then there will not be a recession. The GNP and taxes are both going up. Therefore, the unemployment rate is not going up.

  15. Show that $ P\leftrightarrow Q$ is equivalent to $ \left( P\wedge Q\right) \vee \left( \lnot P\wedge \lnot Q\right) $

  16. Show that $ \left( P\rightarrow Q\right) \wedge \left( Q\rightarrow R\right) $ is equivalent to $ (P \rightarrow R) \wedge [(P \leftrightarrow Q) \vee (R \leftrightarrow Q)]$

  17. Find a formula involving only the connectives $ \lnot $ and $ % \rightarrow $ that is equivalent to $ P \leftrightarrow Q.$

  18. Which of the following formulas are equivalent?

    1. $ P\rightarrow \left( Q\rightarrow R\right) $

    2. $ \left( P\rightarrow Q\right) \wedge \left( P\rightarrow R\right) $

    3. $ \left( P\wedge Q\right) \rightarrow R$

  19. Identify the premises and conclusions of the following deductive arguements and analyze their logical forms. Is the argument valid?

    1. Jane and Pete won't both win the math prize. Pete will win either the math prize or the chemistry prize. Jane will win the math prize. Therefore, Pete will win the chemistry prize.

    2. Either John or Bill is telling the truth. Either Sam or Bill is lying. Therefore, either John is telling the truth or Sam is lying.


next up previous
Next: About this document ...
Jennifer Anne Thacher 2008-09-02
Document Actions