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 xyR(x, y) yxR(x, y).
b. Give a deduction showing that
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-structures N , we have
N
if and only if M N
=
(This pro
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 answers.
Use L = cfw_0, 1, +.
T F x F reeV ar(x + 1 = 1
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 answers.
T F Every embedding is a homomorphism.
T F The
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 given by
g(1) = 3 g(2) = 1
g(3) = 3 g(4) = 7 g(5) = 5 g
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 linear orderings, a sentence
saying that for any two el
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 answers. 7
T Once we prove compactness, the notions of elem
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 justify your answers. I
@ F There is I g Sentp which is con
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_
cfw_
(P C)
(Contr)
Equality Rules:
t=u
t
x
u
x
Ex
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 your answers.
T F There is SentP which is consistent, bu
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 induction on formulas. Write down the statement proved by
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 to soundness. That is, give an example of SentL and Sen
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:
a. Show that the following are equivalent for 1 , 2 Sen