CMPSCI 250: Introduction to Computation

David Mix Barrington

Fall, 2010

Questions and Answers on Homework Assignment #2

HW#2 posted Thursday 23 September 2010

HW#2 due on paper in class, Thursday 7 October 2010

Question text is in black, answers in blue.

• Question 2.1, posted 5 October: Can you tell me again how to set up a quantifier proof? Do I have premises and a conclusion as in the propositional proofs? When Rosen says "therefore" in 73-14, is he separating the premises from the conclusion?

  You're right about the "therefore" -- each of those problems has one or more premises and a conclusion, and he uses the "therefore" to introduce the conclusion. You can organize a quantifier proof using any of the proof methods for propositional proofs. For example, in a direct proof you assume the premise (or premises) P and derive the conclusion C. In a proof by contradiction you assume P ∧ ¬C and derive 0. In an indirect (contrapositive) proof you assume ¬C and derive ¬P.

  Once you've decided what statements you're assuming and what you're deriving, you use the form of those statements to see which of the four quantifier rules you can use: Instantiation for a ∃ in the premise, Existence for a ∃ in the conclusion, Specification for a ∀ in the premise, and Generalization for a ∀ in the conclusion. Generalization has an additional structure, in that you say "let x be arbitrary" at the start, derive some statement P(x) about x, and then conclude ∀x:P(x).

• Question 2.2, posted 5 October: In Rosen #120-30, is an example good enough to prove that A × B ≠ B × A?

• No, the statement that Rosen wants you to prove is about arbitrary sets A and B -- he means ∀A:∀B: [(A ≠ ∅) ∧ (B ≠ ∅)] → [(A × B ≠ B × A) ∨ (A = B)]. Since this is a ∀ statement, you're going to need Generalization. An example would suffice to prove a ∃ statement by Existence.

• Question 2.3, posted 5 October: I don't understand how to multiply one set by another, in Problem 2.1.1 for example.

  You're told that if X and Y are sets, then X × Y is the set of all pairs (a,b), where a is in X and b is in Y. If X is A and B is ∅, what pairs are possible that meet those conditions? The answer is the set A × ∅.

• Question 2.4, posted 5 October: What kind of a proof are you looking for in #85-22, with the socks? I can see that all the possible ways to do it lead to two blacks or two blues, but is that a proof?

  It can be -- if you give a convincing argument that you've covered all the cases, and your conclusion is true in all of them, that's a Proof By Cases. It may be easier to relate your proof to the Chapter 1 proofs if you consider "Sock #1 is blue", "Sock #2 is blue", and "Sock #3 is blue" to be three boolean variables, and show that in all eight settings of those three variables, your conclusion is true. Or you might get a simpler case breakdown that groups some of those eight cases together.

• Question 2.5, posted 5 October: In #48-32 (a), for example, I could see making the type of my variable either "dogs" or "animals". If I use "dogs" my statement is simpler -- is that ok?

  Yes, since Rosen doesn't tell you what type to use, you have your choice and you can make the statement simpler as you say. It's probably worth using "animals" for at least one of the parts, so that you see how to negate and re-express a statement such as "for any animal x, if x is a koala, then x can climb". Note that each part of this problem needs three answers -- a symbolic version of the original statement, a symbolic version of the negation, and an English version of the negation.

• Question 2.6, posted 5 October: In #147-20, I can give a proof in an English paragraph, but do I have to use symbols and the proof rules?

  Yes, in this problem I would like you to use the quantifier proof rules because the premise and conclusion are quantified statements and it's a good simple example of how to use them. Note that I give you a definition of "strictly increasing" which you can easily adapt to get one for "strictly decreasing". Also note that you are asked to prove an "if and only if" statement -- the way I would do this is to prove one direction by direct proof (for example, assume that f is strictly decreasing and prove that g is strictly increasing) and then substitute other functions for f and g to get the second direction from the first.