MATH1090A Problem Set No 4 November–December 2020
Lassonde School of Engineering
Dept. of EECS
Professor G. Tourlakis
MATH1090 A. Problem Set No 4 Posted: Nov. 24, 2020
Due: Dec. 8, 2020; by 3:00pm, in eClass, “Assignment #4”
Q: How do I submit? A:
(1) Submission must be ONLY ONE file
(2) Accepted File Types: PDF, RTF, MS WORD,
ZIP
(3) Deadline is strict, electronically limited.
(4) MAXIMUM file size = 10MB
It is worth remembering (from the course outline):
The homework must be each individual’s own work. While consultations
with the instructor, tutor, and among students, are part of the learning
process and are encouraged, nevertheless, at the end of all this consultation each student will have to produce an individual report rather than a copy
(full or partial) of somebody else’s report.
The concept of “late assignments” does not exist in this course. Page 1 G. Tourlakis
MATH1090A Problem Set No 4 November–December 2020
In what follows, “give a proof of ⊢ A” or “show ⊢ A” means to give
an Equational or Hilbert-style proof of A, unless some other proof style is
required (e.g., Resolution).
Annotation is always required!
Do the following problems (5 MARKS/Each).
1. Prove using soundness (Required):
(∀x)(A ∨ B) → (∀x)A ∨ (∀x)B
2. Prove using soundness (Required):
(∀x)A → (∀x)B (∀x)(A → B).
3. Use the ∃ elimination technique —Required— to show
⊢ (∃x)(A ∧ B) → (∃x)(A → B).
4. Use the ∃ elimination technique —Required; and ping-pong if/where
needed— to show ⊢ (∃x)(A ≡ ¬A) ≡ ⊥.
Do NOT use an Equational proof NOR WL for the above Question (0
marks for such solutions). 5.
(3 MARKS) Prove ⊢ (∀x)(∀y)x = y → (∀y)y = y.
(2 MARKS) Also explain precisely why the above is NOT an instance of Ax2.
Page 2 G. Tourlakis