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日期：2021-10-23 08:25

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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