CS 2603 Homework 6 — Predicates

Due Friday, Mar 25, 2011

NOTE: This assignment is due by 5:00 pm on Friday, March 25, 2011. As I will not be in my office before 5:00 pm that day, please turn in your assignment by giving it to me ahead of time during my office hours or by sliding it under my office door. Do not send assignments to me through email or leave them in my departmental mail box.

1. Motivation

Proving statements and equations of predicate calculus are fundamental processes of formal logic. To understand these processes, it is import to gain experience with them.

2. Goal

The goal of this assignment is to give you experience with proofs of statements and equations that use predicate calculus.

3. Assignment

  1. Prove each of the following statements and write your proof in the format for natural deduction that we have used in class.

    1. ∀x.(f(x) ∧ g(x)), ∀y.(h(y) ∧ i(y)), ∀z.(j(z) ∧ k(z)) ⊢ ∀x(f(x) ∧ (h(x) ∧ k(x)))
    2. ∀x.(p(x) ∧ ∀y.(q(y) ∧ r(x))) ⊢ ∀x.((r(x) ∨ (∃y.s(y))) ∨ u(x))
    3. ∀x.(f(x) → g(x)), ¬∃y.g(y) ⊢ ¬∃x.f(x)   [For this proof, you may not use Modus Tolens.]

  2. Prove each of the following equations and write your proof in the format for equational reasoning that we have used in class.

    1. ∀x.((p(x) ∧ q(x)) ∨ q(x)) = ∀x.q(x)   [For this proof, you may not use ∨ Absorption.]
    2. ∀x.p(x) ∧ ∀y.(p(y) →q(y)) = ∀z.(p(z) ∧ q(z))
    3. ∃x.f(x) ∨ ∃y.g(y) = ¬∀x.(¬f(x) ∧ ¬g(x))

4. Important Notes on this Assignment

  1. NEW NOTE: The statements and equations you are to prove in this assignment may be similar to those you have seen previously in this course. You are free to use the corresponding previous proofs as models for these proofs.

  2. Warnings and Cautions

    The following words of advice come directly from Prof Page who has taught this course many times. Please pay them great heed!

    Warning! For almost all people, the most effective way to work on proofs is to distribute the work over several days. Work a couple hours a day, every day, on the problems. When you’ve worked a good while on one of the problems and find yourself stuck, try another problem. Eventually, problem by problem, you will discover a key that leads to a solution. It may seem that key comes to you suddenly, but somehow the hard work invested before the epiphany gradually builds the picture in your mind until the solution pops out. If you start working on this homework assignment the day before it’s due, you probably won’t finish it. That means you will have missed one of your only real opportunities to prepare for the examinations in this course.

    Importance of Finding Your Own Keys. Each problem that you fail to invest enough time in to find the key yourself reduces, substantially, your chances of passing the exams in this course. I advise you not to discuss a problem with someone else until you have found the key. If you have questions about the material, ask the instructor or the assistant. Email should work well for this. I don’t regard discussing the problems with others as cheating, as long as can explain to me whatever you turn in, but I want you to know that the more you rely on such discussions, the less likely it is that you will succeed on the exams.

    Important Alert! Homework problems provide your only real opportunity to study for exams. Studying the night before the exam will be of almost no use at all. If you have difficulties, come to see the Instructor or Teaching Assistant during office hours.

    How Much Time Will This Homework Take? Some of the problems may take hours to complete. Others, only a few minutes. Some problems may seem difficult to you that seem easy to others, and vice versa. There is no way to predict when or how the insights you will need to solve these problems will come to you. Also, expect some frustration in the process of using the proof checker. Just as in programming, you have to get all the required characters in the right order. All the commas, parentheses, case-sensitive names, etc have to be right. It can easily take an hour to get an already correct proof pushed through the proof checker.

  3. Adding parentheses around an entire statement creates a new statement with the same truth value as the original statement. Likewise, removing parentheses from around an entire statement creates a new statement with the same truth value as the original statement. Therefore, you may add parentheses to or remove parentheses from entire statements within your proofs as you see fit. However, inserting or removing parentheses within a larger statement may change its truth value and is, therefore, not allowed within these proofs. (We will talk later about when we can insert or remove parentheses within statements.)

    For example, if you find yourself at some point in your proof with the statement "X ∧ Y ∨ Z," but you need "(X ∧ Y ∨ Z)" instead, it would be fine to add the parentheses and proceed. Similarly, if you have "(X ∧ Y ∨ Z)," but you need "X ∧ Y ∨ Z" instead, you may remove the parentheses and proceed. However, if you have "X ∧ Y ∨ Z," but you need "X ∧ (Y ∨ Z)" or "(X ∧ Y) ∨ Z" instead, you are not allowed to add the parentheses. Similarly, if you have X ∧ (Y ∨ Z)" or "(X ∧ Y) ∨ Z" but need "X ∧ Y ∨ Z" instead, you are not allowed to remove the parentheses.

  4. Recall that "¬a" is, by definition, the same as "a → False." Therefore, you may substitute one for the other (either direction) in your proofs, as long as you cite this definition.

  5. Modified Note: Recall that you can use as rules in your proofs equations other than those on your handout sheet, provided that those equations have already been proven (in class or in your homework) and that the proof of the equation does not cite, directly or indirectly, the equation you are proving. An equation is "indirectly cited" if it is cited, directly or indirectly, in the proof an equation you are citing.

  6. Modified Note: Don’t forget that a metavariable (such as ‘a’ or ‘b’ in a rule, whether from Natural Deduction or from Equational Reasoning) can stand for any WFF, not just a single variable. For example, you know that you could use {∧ER} with ∀x.f(x) ∧ ∃y.g(y) to derive ∀x.f(x) by substituting ∀x.f(x) for ‘a’ and ∃y.g(x) for ‘b’ in the rule. However, don’t forget that you could use {∧ER} with ∀x.f(x) ∧ (∃y.g(y) ∧ (¬∃z.h(z) ∧ (¬∀w.i(w) ∨ (j(x) → k(y)) ∨ l(z)) ∧ m(a))) to derive ∀x.f(x) as well, simply by substituting ∀x.f(x) for ‘a’ and ∃y.g(y) ∧ (¬∃z.h(z) ∧ (¬∀w.i(w) ∨ (j(x) → k(y)) ∨ l(z)) ∧ m(a)) for ‘b’ in the rule.

5. What to Turn In

You will turn in a typed or neatly written hard copy of your homework that shows all the proofs above in either natural deduction or equational reasoning format as appropriate. You will not need to submit anything electronic for this homework.