Math 200, homework 1 due 9/12/02

Since a number of you have been unable to obtain the book yet, I have transcribed the following problems from the text. I won't do this often, so get a copy of the book as soon as you can.

- (2)
- For each of the following, state whether it is a proposition, with
a brief explanation. If you believe that a particular case is borderline,
provide brief pros and cons for whether it should be considered a
proposition. For those which are propositions, determine which are true and
which are false, if possible.
- 10 is a prime number.
- Are there any even prime numbers?
- Turn off that music or I'll scream.
- Life is good.
- 3+5.
- The number is bigger than 4.
- Benjamin Franklin had many friends.
- The Chicago Cubs will win the World Series in the year 2106.
- I like olives but not very much.
- Goldbach's conjecture is true. (This was described in Chapter 1.)

- (3)
- Determine whether each of the following is a tautology, a
contradiction, or neither. If you can determine answers by commonsense
logic, do so; otherwise, construct truth tables.

- (4)
- Determine whether each of the following pairs of statements are
propropositionally equvalent to each other.
If you can determine answers by commonsense logic, do so; otherwise,
construct truth tables.
- and
- and
- and
- and
- and
- and
- and
- and
- and
- and
- and

- (8)
- Recall the discussion of the inclusive or and the exclusive or.
Let the symbol
represent the latter.
- Construct the truth table for .
- Write a statement using our five basic connectives that is equivalent to .
- Write a statement using only the connectives , , and that is equivalent to .
- Make up an English sentence in which you feel the word ``or'' should be interpreted inclusively.
- Make up an English sentence in which you feel the word ``or'' should be interpreted exclusively.
- Make up an English sentence in which you feel the word ``or'' can be interpreted either way.

- (10)
- For each of the following statements, introduce a propositional
variable for each of its atomic substatements, and then use these variables
and connectives to write the most accurate symbolic translation of the
original statement.
- I need to go to Oxnard and Lompoc.
- If a number is even and bigger than 2, it's not prime.
- You're damned if you do and damned if you don't.
- If you order from the dinner menu, you get a soup or a salad, an entree, and a beverage or dessert. (Be careful with the word ``or'' in this one.)
- If it doesn't rain in the next week, we won't have vegetables or flowers, but if it does, we'll at least have flowers.
- No shoes, no shirt, no service. (Of course, this is a highly abbreviated sentence. You have to interpret it properly.)
- Men or women may apply for this job. (Be careful; this one's a bit tricky.)

- (11)
- (a)
- If a symbolic statement has just one propositional variable (say ), how many lines are in its truth table?
- (b)
- How many different possible truth functions are there for such a statement? That is, in how many ways can the output column of such a truth table be filled in? Explain.
- *(c)
- Repeat parts (a) and (b) for a symbolic statement with two propositional variables and . Explain.
- *(d)
- On the basis of the previous parts of this problem, make conjectures that generalize them to a symbolic statement with an arbitrary number of propositional variables.

Scott Sutherland 2002-09-07