CMPSCI 250: Introduction to Computation

David Mix Barrington

Spring, 2012

Questions and Answers on Homework Assignment #1

HW#1 posted Tuesday 24 January 2012

HW#1 due on paper in class, Friday 3 February 2012

Question text is in black, answers in blue.

• Question 1.1, posted 31 January: For 1.6.5, do we have to have spaces between each pair of bits?

  Yes.

  My first idea is to make a two-dimensional array of booleans, so that I have an entry for each of the k bits in each of the 2^k lines. Then I have to print the array out. Does that make sense?

  You could do it that way, but there's no good reason to have the entire array stored inside your program at once. If you can figure out the first line of your table, then print it, then change it to be the second line, then print that, and so forth, you get the correct output but you don't need all the storage.

  You could even just calculate each bit as you need it, but since Java prints one line at a time it is probably easier to construct the line of text as a String and then print it.

  Aren't the lines just the numbers from 0 through 2^k - 1 in binary notation, except for the spaces?

  Yes. You may be able to use that fact. You are writing in real Java, so you have access to all kinds of methods in classes like Integer if you want to use them. But it is possible to come up with a correct program that doesn't use anything fancy.

  You say "program", but can I just write a static void method that takes an int k as argument and prints the table for that k as a side effect?

  Sure. Or you could have multiple methods if you want.

• Question 1.2, posted 31 January: In Problem 1.5.2, I can't find part (d).

  Part (d) of Problem 1.5.2 is to answer 1.5.2 for the statement in part (d) of Problem 1.5.1. I think you may be looking at Exercise 1.5.2, which is not assigned for the homework because the answer is in the back of the book. Every regular section has both Exercises and Problems -- make sure that you are doing Problems for the homework.

• Question 1.3, posted 31 January: In Problem 1.2.7, should I be writing a boolean method or a method that lists the elements of the language?

  Write a boolean method. A "decision procedure" for a language always inputs a string and then outputs a boolean, which is true if the input string is in the language and false if it is not.

• Question 1.4, posted 31 January: In Problem 1.5.2, how do I tell whether these are "valid set identities"? I believe I have the correct compound propositions from 1.5.1.

  The set statement is a valid set identity if it is true for all possible sets and all possible elements. This is true if and only if the corresponding compound proposition is a tautology (true for any truth values of the atomic propositions x ∈ A, x ∈ B, and X ∈ C). You can test whether you have a tautology by the method of truth tables from Section 1.6. You may be able to argue more simply that you have a tautology. If you don't have a tautology, that is because at least one line of the truth table makes the compound proposition false. To answer 1.5.2, you need to find examples of sets A, B, and C and of an element x where this line of the truth table applies.

• Question 1.5, posted 2 February: In Problem 1.6.5, does it matter which order the rows come in, except for the first and the last? I could have them in order of increasing number of ones, so for k = 3:

  
   0 0 0
   0 0 1
   0 1 0
   1 0 0
   0 1 1
   1 0 1
   1 1 0
   1 1 1
  

  or in the order you use in your truth tables in Section 1.6:

  
   0 0 0
   0 0 1
   0 1 0
   0 1 1
   1 0 0
   1 0 1
   1 1 0
   1 1 1
  

  Would I get points taken off for the wrong order?

  As you say, the problem doesn't specify an order so it would be unfair to take points off. Your second order is much easier to produce, especially if you follow my hint about Integer.toBinaryString, and I prefer it. Among other things it is the alphabetical order on the rows.

  Remember that you do need the spaces between the numbers, and that you need leading zeros so that there are k numbers in each row.

Last modified 2 February 2012