York University. Faculty of Science and Engineering MATH 1090, Section M Final Examination, April NAME (print, in ink): Instructions, remarks:

Size: px

Start display at page:

Download "York University. Faculty of Science and Engineering MATH 1090, Section M Final Examination, April NAME (print, in ink): Instructions, remarks:"

Brook Eaton
5 years ago
Views:

1 York University Faculty of Science and Engineering MATH 1090, Section M ination, NAME (print, in ink): (Family name) (Given name) Instructions, remarks: 1. In general, carefully read all instructions in a problem before beginning it. Make sure you have problems 1 through You have 180 minutes. If you run out of space, continue on the backs of pages. Do not waste your time giving more detail than is asked for. 3. Note: Everywhere in this exam, unless otherwise indicated, the word prove in a problem means give a syntactic proof of. You are NOT allowed to use Completeness (i.e., tautological implication, i.e., Post s theorem) in ANY step of ANY proof on this exam. Proofs can be either Hilbert-style or equational or a mix, unless one type or the other is required in the instructions. (And, as we know, bullet proofs are always equational.) 4. Unless otherwise stated in a problem, it is understood that in solving a problem, you may use any result from an earlier problem on this exam or from an earlier part of the same problem, EVEN IF you did not prove the earlier result. 5. The marks assigned to a problem do NOT necessarily correspond to its difficulty. There are 127 marks available on this exam, but it will be marked out of 100. Problem Marks Name 2 Total 127

2 Page 1 1. Give a bullet proof that p (q p) p q. (I guess I know of only one way to start the proof or at least only one best way according to our general strategy for approaching bullet proofs the troublesome connective here is find a way to get rid of it. Bullet always means no ping-pong, no D.T.) 2. Give a bullet proof that, for any wffs A and B, ( A B) (A B).

3 Page 2 3. One of the following wffs A, B is a theorem; the other is not. A is ( (p q) q ) p. B is ( (p q) p ) q. (a) Give a bullet proof of the theorem. (The Deduction Theorem is NOT allowed in this problem. You may use any theorems on your lists; you may also use Shunting: (C D) E C (D E). You may not use other theorems than the ones mentioned above unless you prove them first.) (b) Explain carefully, using and naming facts from the course, how we know that the other wff is not a theorem.

4 Page 3 4. GIVE A BULLET PROOF that the wff below is a theorem. In your proof, you may use any theorem on your lists. (p r) (p q) q (r p) (Getting rid of the arrows in a SMART way gives a very short proof. Note also that the first disjunction has a in it, and the third does not. That might remind you of something.)

5 5. Several unrelated problems. Page 4 (a) State the Completeness Theorem, as given in class. (It can be stated the same way for predicate logic as for boolean logic, with the same symbols. So you don t have to say which logic you are talking about; it works for both.) (b) State the general form of the Deduction Theorem, as given in class. (Again, the same version works in both boolean and predicate logic.) (c) State the general form of the Soundness Theorem, as given in class. (Same comment as in parts (a), (b).)

6 Page 5 (d) In each case below, finish the sentence in one of two ways. Either write undefined., or not defined., or write out the wff that results from the substitution (write it out without [ := ] notation, of course). You may show work or explain, or not, as you prefer. Correct answers without explanation will get full marks. Wrong answers with no explanation get 0. Wrong answers with some relevant explanation MAY get part marks. w, x, y and z here are really the variables they seem to be. They are four different variables. There is a superfluous pair of parentheses in part iv. below (and, later, in part (e)), to clarify what is being asked. i. ( ( x)(( y) (y = z)) ) [z := y] is ii. ( ( x)(( y) (y = z)) ) [y := x] is iii. ( ( x)(( y) (y = z)) ) [y := w] is iv. ((y = w) [w := y]) [y := w] is (e) True or false?: For all wffs A and all object variables x, y, (A [x := y]) [y := x] is A. Justify your answer.

7 Page 6 6. Let A and B be wffs. Give a BULLET proof that ( x) (A B) ( x) A ( x) B. You may use any theorem on the lists that you have. 7. Using any method you like, and anything on your lists, prove syntactically that, for any wff A, ( x)a ( x)a.

8 Page 7 8. (a) Give a Hilbert-style proof that (x = y y = x). (b) Give a VERY short Hilbert proof that, for any boolean wffs A, B, D, (D (A B)) (D (B A)) is a theorem. Explain every little thing in your proof, using standard Hilbert format.

9 Page 8 9. (a) A very short, easy proof of this fact was posted on the course page: If x dnof in a wff D, then ( x) D D. Now take this fact as GIVEN, and use it to prove that if x dnof in D then ( x) D D. (b) Suppose that x does not occur in ( y)a. Prove that it is. The following wff is then a theorem. ( x) ( A[y := x] ( y) A ) (But FIRST: See bottom of page.) Note: A bullet proof works. First use our favourite way to get rid of an arrow. Then use Problem 6 even if you didn t prove it, then part (a) above, to simplify something. Finally: Look again at the first sentence of 9.(b). Why is dno there rather than dnof?

10 Page Give a bullet proof that, for all wffs A, B, ( x)(a ( x)b) (( x)a ( x)b) is a theorem. (There is a very short proof.)

11 Page Prove the following English statement. (Using the Deduction Theorem is strongly recommended here.) Given any wffs P, Q, R, if y dnof in R and x dnof in Q, then ( x)( R ( y)(q P ) ) ( y)( Q ( x)(r P ) ). You may NOT use Ax3 in your proof. I want a proof that uses our basic techniques but does not use Ax3. Before jumping into the proof, THINK about what combination of standard techniques will lead to a proof. If your proof used Ax3, go back and find a new proof.

12 Page Use metatheorem to prove that { ( x)(a B), ( x)a } ( x)b. No credit will be given for a proof that does not use (This was an example in our text, understanding which I recommended as good exam preparation.) Write up the proof as such proofs are written in our text, and in accordance with the wording of on the sheets you have at this exam.

13 Page Show that ( x)(a B) A ( x)b is, in general, NOT a theorem. In particular, find specific wffs A and B such that the above conditional is NOT a theorem. (Look at if you did not already think of doing that.) Explain in detail, using facts from the course, how you KNOW that the above conditional is not a theorem, for your A and B. 14. Let φ be a 2-ary predicate symbol. Find a model in which the following wff interprets as true: ( x)( y)(φ(x, y)) ( x)( y)( z)( ( φ(x, y) φ(y, z) ) φ(x, z) ) ( x)( φ(x, x)) Don t explain unless you want to. Just name the domain, D, and the interpretation of φ(, ).

MATH 1090 Problem Set #3 Solutions March York University

MATH 1090 Problem Set #3 Solutions March York University York University Faculties of Science and Engineering, Arts, Atkinson MATH 1090. Problem Set #3 Solutions Section M 1. Use Resolution (possibly in combination with the Deduction Theorem, Implication as