MAT 3361, INTRODUCTION TO MATHEMATICAL LOGIC, Fall 2004
Handout 2: Quantier Problems
Problem 1 Prove the following in natural deduction:
(a) Q x P (x) x (Q P (x) assume that x does not occur in Q.
(b) x P (x) y P (y).
(c) x P (x) x Q(x) x (P (x) Q(x).
(d)
MAT 3361, INTRODUCTION TO MATHEMATICAL LOGIC, Fall 2004
Handout 1: Rules of Fitch-style natural deduction
Conjunction introduction (I)
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MAT 3361, INTRODUCTION TO MATHEMATICAL LOGIC,
Fall 2004
Answer:
1
Answers to the Midterm
AB
2
Problem 1. Prove the following in natural deduction, using your
choice of Fitch or Prawitz style:
B
3
C
4
(a)
5
Answer:
A
B
7
A C.
A
6
(A B) C
E, 2, 6
A
E, 7
1
(
MAT 3361, INTRODUCTION TO MATHEMATICAL LOGIC, Fall 2004
Lecture Notes: Analytic Tableaux
Peter Selinger
These notes are based on Raymond M. Smullyan, First-order logic. Dover Publications, New York 1968.
1 Analytic Tableaux
Denition. A signed formula is a
2 Soundness and Completeness for Analytic Tableaux
Recall that we have called a branch of a tableau complete if every formula on
it has been used. With our convention on using the letters and for signed
formulas, we may express this more precisely:
A bran
1. Give natural deduction proofs of the following. Make sure that you label each step by the
name of the rule you use and that you use correct numbering for cancelled hypotheses.
(a) [2 points]
( )
Answer.
E
E
I
I
( )
(b) [3 points] ,
( ) ( )
Answer.
MAT3361 Midterm 2 study sheet
The format of the exam is the same as the closed midterm. The main topics are:
Predicate Logic: You need to understand the syntax (terms, formulas, variables, quantiers)
and the rules governing the syntax; in particular you n
MAT 3361, INTRODUCTION TO MATHEMATICAL LOGIC, Fall 2004
Problem 41
[A]2
A A
Answers to Problem Set 2:
Handout 1, 11, 21, 31, 41, 51 (using Prawitz style natural deduction),
12, 22, 32, 42, 52 (using analytic tableaux)
[(A A)]1
( I2 )
A
(I)
A A
[(A A)]1
(
MAT3361 Introduction to Mathematical Logic
Midterm Test #2 Solutions
1. Give natural deduction proofs of the following. Make sure that you label each step by the
name of the rule you use and that you use correct numbering for cancelled hypotheses.
(a) [2
MAT3361 Midterm #1 study sheet
The exam is closed book. The material consists mainly of the following:
1. Natural Deduction proofs for propositional logic. On the exam you will get a list of all the
ND rules, so you dont have to memorize those. However, y
MAT 3361, INTRODUCTION TO MATHEMATICAL LOGIC, Fall 2004
5
Problem 1 (a) (A B)
1
2
(A B)
E, 1
E, 6, 7
9
A
(A B)
(c) A B
4
AB
I, 3, 4
6
E, 1, 5
E, 2, 35, 68
I, 29
(A B)
B
5
E, 3, 4
10
( A B)
3
B
8
A B
B
7
Peter Selinger
6
Answers to Problem Set 1: Handout 1