100% satisfaction guarantee Immediately available after payment Both online and in PDF No strings attached
logo-home
Summary UCL ECONS MATH0050 Complete Study Notes £9.99
Add to cart

Summary

Summary UCL ECONS MATH0050 Complete Study Notes

1 review
 14 purchases

• 48-pages digital & handwritten study notes for MATH0050 Logic • Seller achieved high 1st (>90%) • Organized into 3 sections: Section A (common exam questions & additional notes), Section B (list of all definitions & proofs across chapters), Section C (actual definitions & proofs (handwri...

[Show more]

Preview 5 out of 48  pages

  • April 26, 2020
  • 48
  • 2018/2019
  • Summary
All documents for this subject (54)

1  review

review-writer-avatar

By: freyaliiny • 2 year ago

avatar-seller
1stclassecons


MATH0050 Logic
Table of Contents
Section A: Common questions & additional notes
Chapter 1 Language
Chapter 2: Propositional Logic
Chapter 3: First Order Predicate Logic
Chapter 4: Computability
Section B: List of definitions and proofs
Chapter 1 Language
Chapter 2 Propositional Logic
Chapter 3 First Order Predicate Logic
Chapter 4 Computability
Section C: Definitions and proofs
Chapter 1, 2 & 4: Definitions & proofs (handwritten)
Comparison: Chapter 1 vs Chapter 2
Comparison: Chapter 2 (Syntactic vs Semantic)
Comparison: Chapter 2 vs Chapter 3

,Section A: Common questions & additional notes
Chapter 1 Language
1. Calculate degree & weight

Degree Weight
Variable symbol 0 -1
n-ary predicate symbol 0 n-1
¬ Add 1 0
⇒ Add 2 +1
Ɐ Add 1 +1


2. Determine whether if it is a formula / Determine if string is an element of L
Defining properties of
formulae

Check weight 1. If weight ≠ -1 then conclude not formula
2. If weight = -1, need to check with definition of “formula”

Proper initial segment Check cumulative weight and see if we reach a negative weight before
the string ends



3. Covert into a formula in L
αvβ “or” (¬α) ⇒ β If does not have α, we must
have β
α˄β “and” ¬ (α ⇒ (¬β)) Not possible to have α and
not have β
α⇔β “if and only if” (α ⇒ β) ˄ (β ⇒ α)
Ǝxα Exists x such that α is satisfied ¬ Ɐx¬ α Not possible that for all x, α
is not satisfied

, 4. Semantic equivalent vs Semantic tableau vs Truth table


Convert into formula in L Semantic tableaux
Truth table
(semantically equivalent)
αvβ (¬α) ⇒ β
α β αvβ
0 0 0
1 0 1
0 1 1
1 1 1

α˄β ¬ (α ⇒ (¬β))
α β α˄β
0 0 0
1 0 0
0 1 0
1 1 1


α⇔β (α ⇒ β) ˄ (β ⇒ α)
α β α⇔β
0 0 1
1 0 0
0 1 0
1 1 1




Ǝxα ¬ Ɐx¬ α N/A N/A
α⇒β N/A
α β α⇒β
0 0 1
1 0 0
0 1 1
1 1 1

¬¬ α N/A
α ¬α ¬¬ α
0 1 0
1 0 1

,Chapter 2: Propositional Logic
1. Use semantic tableaux method to prove if semantic implication holds / tautology
Question Solution
(note the position of the ⊨ symbol)
Semantic We form a semantic tableau starting from
implication the propositions:

*3 separate propositions
Semantic We form a semantic tableau starting from
implication /
tautology
If there are open branches, must describe every valuation which it fails to hold:
By considering the open branch, we may describe a type of valuation v for which the given proposition fails to
be true. For a valuation v, if v(α)=0/1, v(β)=0/1 and v(γ)=0/1, then v(……) = 0

2. Use truth table to prove tautology
“The column corresponds to π contains only ones”

3. Direct proof for syntactic implication
a. α ⇒ α
b. ¬¬ α ⇒ α (use α ⇒ α as ‘assumed’)
c. α ⇒ ¬¬ α (use ¬¬ α ⇒ α as ‘assumed)

4. Use deduction theorem to prove syntactic implication
“By Deduction theorem, it suffices to give a proof of … “
“A further application of deduction theorem indicates that it suffices to give a proof of…”

5. Some theorems that you can assume
a. ¬¬ α ⇒ α
b. α ⇒ ¬¬ α
c. α ⇒ α

6. Axioms:
a. Axiom 1: want to add something in front of β
b. Axiom 3: change of negations
c. Modus ponens: cancel something in front of β

, Chapter 3: First Order Predicate Logic
1. Describe a theory

Π and Ω Sentences
Graphs Π = {∼}
Ω=φ




Graphs Π = {∼ , =}
(when add equality Ω=φ Add:
symbol) Reflexibity, symmetry,
transistivity
(about the = sign)
Substituvity sentences
(interaction of = and ~ )


Posets Π = { = , ≤}
Ω=φ

About the ≤ sign



Reflexibity,
symmetry,
transistivity
(about the = sign)

Substituvity sentences
(interaction of = and ≤)


Groups Π={=}
Ω={.,E}

About the . and E




Reflexibity, symmetry,
transistivity
(about the = sign)


Substituvity sentences
(interaction of = and . )

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

Quick and easy check-out

You can quickly pay through credit card for the summaries. There is no membership needed.

Focus on what matters

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller 1stclassecons. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for £9.99. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

65040 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy revision notes and other study material for 15 years now

Start selling
£9.99  14x  sold
  • (1)
Add to cart
Added