Assignment 3.  (1/18, due 1/23)

Read section 2.1.  Do problems 4, 6, 8, 10 on pages 36-36.  (You have to read problem 9 in order to do 10.)  Write only the answers: for 4 and 8, write T or F for each part.  For 6, write the truth table.  For 10, write an English sentence for each part (with capital letter and period).  You may drop the words ``the integer'' from "the integer x is even."  That is, you can assume it has been stated in advance that the letter x stands for an integer.  The reason I say this is that I don't like the book's answer for 9d (page 482).  They say "The integer x^2 is even is necessary for x to be even."  They did this because the sentence can't begin with x^2, there must be a word in the beginning.  So they moved "the integer" to that position.  But this is actually a bit wrong, because the original sentence said that x was an integer and did not say anything about x^2.  (If x is an integer, then x^2 is also, but if x^2 is an integer, it does not follow that x is an integer, so saying x^2 is an integer is not the same as saying that x is.)  A better way of writing the answer to 9d would be "It is necessary that x^2 be even for x to be even."

Also answer the following question, which is related to the extra question in assignment 2.  If n is a positive integer, let f(n) be the number of intervals remaining when n distinct points are removed from a line.  Find a formula for f(n+1) in terms of f(n), and explain in your own words why the formula is true.  (This type of formula is called a recursion relation.)