CMSC250, Spring 2004 Homework 1 Answers

Due Wednesday, February 4 at the beginning of your discussion section.

  1. Use the following propositions for this problem. Write each of the following statements in symbolic form using the symbols $ \sim $, $ \wedge $, and $ \vee $ and the letters defined above.
    1. John is a basketball player, however he is not a computer science major.
      • Answer: $ b \wedge \sim c$
    2. John is either a hard worker or a basketball player, but not both.
      • Answer: $ (h \wedge \sim b) \vee (\sim h \wedge b)$     OR      $ (h \vee b) \wedge \sim (h \wedge b)$
    3. John is a hard worker, but he is neither a basketball player nor a computer science major.
      • Answer: $ h \wedge \sim b \wedge \sim c$

  2. Use DeMorgan's Laws to write the negations of the following as English sentences.
    1. Mary and Sam are both engineering majors.
      • Answer: Mary is not an engineering major or Sam is not an engineering major.
    2. Morgan is is neither a swimmer nor a runner.
      • Answer: Morgan is either a swimmer or a runner (or both).
    3. Either the weather is nice or the game is cancelled.
      • Answer: The weather is not nice and the game is not cancelled.

  3. Assuming $ x$ is a particular real number,

    1. write the negation of this statement: $ -1 < x \leq 4$.
      • Answer: $ (-1 \geq x) \vee (x > 4)$
    2. Give one possible value that $ x$ could be if the statement `` $ -1 < x \leq 4$'' is false.
      • Answer: Many possible answers. Any number less than or equal to -1 or greater than 4 is fine.

  4. State whether each of the following sentences are statements. If a sentence is a statement, state whether it is true or false.

    1. ``Why is the sky blue?''
      • Answer: Not a statement -- it's a question.
    2. ``My CMSC250 instructor is Jan Plane.''
      • Answer: True statement.
    3. ``This sentence is not a statement.''
      • Answer: False statement.
    4. ``This sentence is both true and false.''
      • Answer: False statement.

  5. Construct a complete a truth table for each of the following statements, and tell whether each is a tautology, contradiction, or neither.
    1. $ (p \wedge \sim q) \wedge (\sim p \vee q)$
      • Answer:
        $ p$ $ q$ $ \sim p$ $ \sim q$ $ p \wedge \sim q$ (X) $ \sim p \vee q$ (Y) (X) $ \wedge $ (Y)
        1 1 0 0 0 1 0
        1 0 0 1 1 0 0
        0 1 1 0 0 1 0
        0 0 1 1 0 1 0



        This statement is a contradiction.

    2. $ \sim p \wedge (q \vee \sim r)$
      • Answer:
        $ p$ $ q$ $ r$ $ \sim p$ $ \sim r$ $ q \vee \sim r$ $ \sim p \wedge (q \vee \sim r)$
        1 1 1 0 0 1 0
        1 1 0 0 1 1 0
        1 0 1 0 0 0 0
        1 0 0 0 1 1 0
        0 1 1 1 0 1 1
        0 1 0 1 1 1 1
        0 0 1 1 0 0 0
        0 0 0 1 1 1 1



        This statement is a neither a tautology nor a contradiction.

    3. $ (p \wedge \sim q) \vee (p \wedge q) \vee \sim p$
      • Answer:



        $ p$ $ q$ $ \sim p$ $ \sim q$ $ p \wedge \sim q$ (X) $ p \wedge q$ (Y) (X) $ \vee $ (Y) (X) $ \vee $ (Y) $ \vee \sim p$
        1 1 0 0 0 1 1 1
        1 0 0 1 1 0 1 1
        0 1 1 0 0 0 0 1
        0 0 1 1 0 0 0 1



        This statement is a tautology.

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