Math 1090 Homework 1 Math 1090 Homework 1
Solutions

  1. In translating each of the following to symbolic form, the word ``or'' would be replaced by or \veebar . What is the appropriate choice in each case? Explain why your choice is correct.

    1. To take discrete mathematics, you must have taken calculus or a course in computer science.
      Answer: since it is fine if you have taken both.
    2. Dinner for two includes two items from column A or three items from column B.
      Answer: \veebar since if you choose two from A you cannot in addition choose three from B.
    3. When you buy a new car from Acme Motor Company, you get $ 2000 back in cash or a 2% car loan.
      Answer: \veebar since you get one or the other but not both.
    4. School is closed if more than 2 feet of snow falls or if the wind chill is below -50.
      Answer: since school is closed if both happen to be the case.

  2. Determine whether
    ( ~ q (p q))   ~ p
    is a tautology.
    Answer:

    p q ( ~ q (p q)) ~ p
    TTFFTTF
    TFTFFTF
    FTFFTTT
    FFTTTTT

    Yes it is a tautology.

  3. Does p q logically imply (p q)  ~ p ? Justify your answer.
    Answer:

    p q p q (pq) ~ p
    T T T T T F
    T F F F F F
    F T T F T T
    F F T F T T

    Yes. Whenever p q is T, so is (pq)  ~ p.

  4. Use truth tables to show that both
    p (p q)  and  p (p q)
    are equivalent to p.
    Answer:

    p q p (pq) p (p q)
    T T T T T T T T
    T F T T F T T T
    F T F F F F F T
    F F F F F F F F

  5. Determine whether
    (p q) r   and  p (q r )
    are equivalent.
    Answer:
    p q r (pq) r p (q r)
    T T T T T T T T T
    T T F T F F T T F
    T F T T T T T T T
    T F F T F F T T T
    F T T T T T F T T
    F T F T F F F F F
    F F T F T T F T T
    F F F F T F F T T

    Their truth tables are different. They are not equivalent.

    1. Determine whether
      (p q) r   and  p (q r )
      are equivalent.
    2. Is ((p q) r)     (p (q r )) a tautology? Explain.

    Answer:
    (a) They are equivalent. Look at their truth tables:
    p q r (p q)r p (q r)
    TTTTTTTTT
    TTFTFFTFF
    TFTFFTTFF
    TFFFTFTTT
    FTTFFTFFT
    FTFFTFFTF
    FFTTTTFTF
    FFFTFFFFT

    (b) Look at the truth table above. ((p q) r)     (p (q r )) is a tautology.
    It always has truth value T.


File translated from TEX by TTH, version 2.60.
On 21 Sep 2000, 20:12.