Math 39 Homework #1

The thing that counts is not what we know, but the ability to use what we know.
Leo Spears

Quiz

Remember that we have a quiz on Monday 9/28 on Lecture 1.

Reading

Note that all reading assignments are from Hamilton's book, unless Crossley is specifically cited.

Read what we covered in today's class: 1.1, 1.2.
Read what we will cover in the next class: 1.3, 1.4.

When reading 1.1 pay special attention to the following concepts:

  • connectives, statements, statement forms, statement variables, truth values.
When reading 1.2 pay special attention to the following concepts:
  • negation, conjunction, disjunction, conditional, biconditional, statement form, tautology, contradiction, logically implies, logically equivalent.
When reading 1.3 pay special attention to the following concepts:
  • There are 7 propositions and corollaries in this section; paraphrase what they mean? Do you understand why they are true? Why would we be interested in such results?
When reading 1.4 pay special attention to the following concepts:
  • normal form, disjunctive normal form, conjunctive normal form.

Problems

The following problems are due by the beginning of class on Monday 9/28.
  • 1.1: 1, 2a
  • 1.2: 4, 5, 7
Also, solutions to some exercises are listed in the back of the book. One of piece of the Honor Principle for this course is that you are not allowed to look at these solutions until you have finalized the copy of the homework you will submit for a grade.

Finally, please use the following "dictionary" when working on problem 1.1.1. Thanks for your help.

  • A: both numbers are even
  • B: both numbers are odd
  • C: the sum of two numbers is even
  • D: demand is constant
  • H: someone is harbouring him
  • J: Jones is elected party leader
  • M: the murder has left the country
  • P: prices have increased
  • R: Robinson shall leave the cabinet
  • S: Smith shall leave the cabinet
  • T: turnover has increased
  • W: we shall win the election
  • X: x is rational
  • Y: y is an integer
  • Z: z is real
A Final Addendum (made on 9/27):

I received the following message:
For problem 1.1.1 part a, I'm having difficulty completing the problem using your given dictionary. In particular, your definition of T is leading to some problems; since increasing and decreasing are not mutually exclusive, negating T means that turnover has decreased or remained constant, which isn't what the excercise says. Should I modify your dictionary or did you intend for us to find a more complex solution?

The answer:
Good observation...my definition of T was intended to force you to use a negation. But you're right that some modification needs to be made to the dicitonary. There seem to be two choices..change the definition of T or add a new simple statement to the dictionary. Pick one, indicate the modifications you have made on your homwork, and make the corresponding translation of #1.1.1a. Note that there are different reasons to prefer each choice. Include a list of these with your solution of the problem.