Math 125 A Fall 2013
Homework 2: Due Wednesday, September 19
Required Problems
Denition: Let 1 , 2 Sentp . We say that 1 and 2 are semantically equivalent if 1
and 2 for all 1 .
for all 2
Problem 1:
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Silvain Rideau
1091 Evans
Homework 5
Due October 8th
Problem 1 :
1. A sentence is said to be universal if it is of the form x1 . . . xn (x1
Silvain Rideau
1091 Evans
[email protected]
www.normalesup.org/~srideau/eng/teaching
Homework 6
Due October 29th
Problem 1 :
We want to replace the rule (Def ) (which says that (x) (x) alway
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www.normalesup.org/~srideau/eng
Silvain Rideau
1091 Evans
Homework 4
Due October 1st
Problem 1 :
1.
a) In N1 this formula is interpreted as: for all x and y N, x + (y + 1)
Silvain Rideau
1091 Evans
[email protected]
www.normalesup.org/~srideau/eng/teaching
Solutions to Homework 6()
Due October 29th
Problem 1 :
1. By the (3 ) axiom, x holds. Moreover (A B) (B A
Silvain Rideau
1091 Evans
[email protected]
www.normalesup.org/~srideau/eng/teaching
Homework 8
Due November 12th
Problem 1 :
1. Let x, y be in A, t is their lower upper bound in (A, ) if an
Silvain Rideau
1091 Evans
[email protected]
www.normalesup.org/~srideau/eng/teaching
Homework 7
Due November 5th
Problem 1 :
Recall that a universal sentence is a sentence of the form x1 . .
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Silvain Rideau
1091 Evans
Homework 4
Due October 1st
Problem 1 :
Let L be the language consisting of a unary function symbol f , binary func
Silvain Rideau
1091 Evans
[email protected]
www.normalesup.org/~srideau/eng/teaching
Homework 10
Due December 1st
Problem 1 :
Let I = x x and W = xy(x)y)y .
1. Give all t such that (x (I)x)x
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www.normalesup.org/~srideau/eng
Silvain Rideau
1091 Evans
Solutions to Homework 5
Due October 8th
Problem 1 :
1. Let us first prove that for any quantifier free formula (x1
Silvain Rideau
1091 Evans
[email protected]
www.normalesup.org/~srideau/eng/teaching
Homework 9
Due November 19th
Problem 1 :
1. Recall that a nite Boolean algebra is atomic. Hence, if aA ac
Silvain Rideau
[email protected]
www.normalesup.org/~srideau/eng/teaching
1091 Evans
Solutions to homework 10
Due December 1st
Problem 1 :
1. Let
= x (x)x.
There are eight
(x (I)x)x)x (I)x
Silvain Rideau
1091 Evans
[email protected]
www.normalesup.org/~srideau/eng/teaching
Homework 8
Due November 12th
Problem 1 :
Let (A, 0, 1, +, ) be a boolean algebra. For all x and y A, dene
Silvain Rideau
1091 Evans
[email protected]
www.normalesup.org/~srideau/eng/teaching
Homework 9
Due November 19th
Problem 1 :
Let A be a nite Boolean algebra. Let A denote the set of its ato
Math 125 A Fall 2013
Homework 8: Due Monday, November 18
Required Problems
Problem 1: Let L = cfw_R where R is a binary relation symbol, and let x, y V ar be distinct.
a. Give a deduction showing that
Math 125 A Fall 2013
Homework 9: Due Monday, November 24
Required Problems
Problem 1: (a) Suppose that the rule (P) did not have the conditional that y F reeV ar( cfw_x, ),
and give a counter-example
Math 125 A Fall 2013
Homework 6: Due Friday, October 25
Problem 1: Let L = cfw_R where R is a binary relation symbol and let M be a nite L-structure. Show that
there exists SentL such that for all L-s
Math 125 A Fall 2013
Homework 4: Due Friday, October 11
Required Problems
Problem 1: Decide whether the following statements are True or False. Circle the right answer. You dont
need to justify your a
Math 125 A Fall 2013
Homework 5: Due Friday, October 18
Required Problems
Problem 1: Decide whether the following statements are True or False. Circle the right answer. You dont
need to justify your a
Math 125 A Fall 2013
Homework 1: Due Wednesday, September 11
Required Problems
Problem 1: Consider the following generating system. Let U = cfw_1, 2, 3, 4, 5, 6, 7, and F = cfw_f, g, where
g : U U is
Math 125 A Fall 2013
Homework 7: Due Friday, November 1st
Problem 1: Let L = cfw_. Let DLO be the axiom for dense linear orderings without
endpoints. That is, DLO is the conjunction of the axioms for
Math 125 A Fall 2013
Midterm 2: November 4
Name: . . . /30
Problem 1: (10 points) Decide whether the following statements are True or False. Circle the right answer.
You dont need to justify your answ
Math 125 A Fall 2013
Homework 3: Due Wednesday, September 25
Required Problems
Problem 1: Decide whether the following statements are Trueor False. Circle the right answer. You dont . :
need to justi
SYNTACTIC IMPLICATION
Basic Proofs:
if (AssumeL )
t = t for all t T ermL (EqRef l)
Proof Rules:
(EL)
(ER)
(IR)
cfw_
(P C)
cfw_
(I)
cfw_
( I)
( E)
cfw_
(IL)
cfw_ cfw_
cfw_
cfw_
Math 125 A Fall 2013
Homework 3: Due Wednesday, September 25
Required Problems
Problem 1: Decide whether the following statements are True or False. Circle the right answer. You dont
need to justify y
Math 125 A Fall 2013
Homework 10: Due Friday, December 6
Problem 1: In class we showed that if is complete, consistent and contains witnesses, then its term model
M is a model of . The proof was by in
Silvain Rideau
1091 Evans
[email protected]
www.normalesup.org/~srideau/eng/teaching
Homework 7
Due November 5th
Problem 1 :
1. Let us assume T is consistent otherwise T is not consistent ei