关于算法:MATH1090-Problem

MATH1090 Problem Set No2 October 2021
Lassonde School of Engineering
Dept. of EECS
Professor G. Tourlakis
MATH1090 B. Problem Set No2
Posted: Oct. 6, 2021
Due: Oct. 29, 2021; by 2:00pm, in eClass.
Q: How do I submit?
A:
(1) Submission must be a SINGLE standalone
file to eClass. Submission by email is not
accepted.
(2) Accepted File Types: PNG, JPEG, PDF,
RTF, MS WORD, OPEN OFFICE, ZIP
(3) Deadline is strict, electronically limited.
(4) MAXIMUM file size = 10MB
Post’s Theorem use is not allowed in any question
below.

1. (3 MARKS)
   We proved in class that
   ` A ≡ A
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   MATH1090 Problem Set No2 October 2021
   using the“trick”of a Leibniz“mouth”-variable p that does not appear
   in A.
   Prove this again, Equationally, but without using this trick and without
   using Post’s Theorem.
2. (5 MARKS) Prove Equationally that A, B ` A ≡ B.
3. (5 MARKS) Is Statement (1) below True or False and WHY?
   Γ A≡B is equivalent to“Γ A IFF Γ ` B”(1)
   Note that the sub-statement in quotes is a METAstatement. Note also
   that we have two“iff”in (1) above!
    Caution. If a proof style is explicitly required in what follows, then any
   other style used gets 0 marks even if it is correct. 
4. (5 MARKS) Prove Equationally that for any A and B
   A, ¬A ` B ≡ ¬B
5. (4 MARKS) Prove Equationally that ` A → B → A.
6. (4 MARKS) Prove Equationally that A → B ` ¬B → ¬A.
7. (5 MARKS) Prove (choose your favourite: Equational or Hilbert proof)
   that A → B ` (B → C) → A → C.
8. Prove that A → B, C → B ` (A ∨ C) → B.
   two proofs are required:
   • (3 MARKS) One with the Deduction theorem (and a Hilbert-style
   proof; CUT rule allowed in this subquestion).
   • (4 MARKS) One Equational, WITHOUT using the Deduction theorem.
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