MIDTERM 2
Math 114
5/15/2009
Name:
Read all of the following information before starting the exam:
Show all work, clearly and in order, if you want to get full credit. I reserve the right to
take o points if I cannot see how you arrived at your answer (e
MIDTERM 1
Math 114
4/24/2009
Name:
Read all of the following information before starting the exam:
Show all work, clearly and in order, if you want to get full credit. I reserve the right to
take o points if I cannot see how you arrived at your answer (e
MIDTERM 1
Math 114
4/23/2010
Name:
Read all of the following information before starting the exam:
Check your exam to make sure all pages are present.
Show all work, clearly and in order, if you want to get full credit. I reserve the right to
take o poi
Math 114L
Homework 3 Solutions
Spring 2011
2.1.1
1.
2.
3.
4.
5.
6.
dxx 0 or dxx 0 x 0
hxIx I 0
dx2x 0
dxpp2Ix dyy x Iyq Ixq
dx2pdyy xq or perhaps dx2pdyy x y xq
dx2dy2y x
2.1.4
pdxEx Axq dx pphyEy x hyq phypAy x hyqqq
2.1.10
1.
2.
p2pp2dv1p2v1P v1qq p2P v
Math 114L
Homework 2 Solutions
Spring 2011
1.5.1
1.5.1a
p2A1 2A2 2A3 qp2A1 2A2 A3 qp2A1 A2 2A3 qpA1 2A2 2A3 q
1.5.1a
pA1 A2 q 2pA3 pA1 A2 qq
1.5.3
We will prove by induction that if is a w built only from 2 and # and
containing the sentence symbols A, B t
Math 114L
Homework 1 Solutions
Spring 2010
1.1.2
We rst show that there are ws of every length other than 2, 3, 6 with some
examples followed by induction on natural numbers. The sentence symbol A1
is a w of length 1. (A1 ) is a w of length 4. The w (A1 A
Math 114L
Homework 3 Solutions
Spring 2011
Solution to 5b
Let n be the formula
Rx1 x2 Rxn1 xn Rxn x1
and let n be
x1 x2 xn n .
Let = cfw_n | n 1. If A contains a cycle a1 , . . . , an then,
A
n [a1 , . . . , an ]
and therefore
A
Conversely, if A
a1 , . .
Final Exam, Math 114
Due 2:30, Thursday June 9th
You MAY: use your notes, use your textbook, ask general questions of the professor or TA
You MAY NOT: discuss problems with other people, research questions on the internet
1. Prove, using the formal induct
Math 114L
Homework 2 Solutions
Spring 2011
1.7.2
Let be nitely satisable and complete, and let be as given in the problem.
We show by induction on that () = T i .
Base Case: If is a sentence symbol, () = () = T i by the
denition of .
Inductive Case for :
MIDTERM 2
Math 114
5/14/2010
Name:
Read all of the following information before starting the exam:
Show all work, clearly and in order, if you want to get full credit. I reserve the right to
take o points if I cannot see how you arrived at your answer (e