Mathematics 114C, Winter 2017
Yiannis N. Moschovakis
Final examination, March 23, 2017
Name (last name first):
Signature:
There are 15 problems or parts of problems and each of them is worth 8 points, for a
total of 120 points, 20 more than the 100 points
Mathematics 170A HW7 Due Tuesday, February 28, 2012.
Problems 38, 40, 41(a,b,c), 42(a,b) on pages 132-133 and problem 1
on page 184.
I1 . Let X1 and X2 be independent geometric random variables with
parameters p1 and p2 respectively.
(a) Find P (X1 X2 ).
Mathematics 114C, Winter 2017
Yiannis N. Moschovakis
First Midterm, February 7, 2017
Name (last name first):
Signature:
There are 110 points in this test and 100 points count for a perfect score, so you
can skip a part or afford to make a small mistake.
T
Math 114C, Winter 2017, Solutions to HW #6
x4B.5. Does there exist a total, recursive function f (e, m) such that for all
e, m,
Wf (e,m) = cfw_x + y | x We and y Wm ?
You must prove your answer.
Solution. It is true: we set
R(e, m, t) (x, y)[t = x + y & x
Mathematics 170A HW8 Due Tuesday, March 6, 2012.
Problems 5,6,11,12 on pages 186-188.
J1 . A point (X, Y ) is chosen uniformly from the unit square [0, 1]
[0, 1]. Find the CDF and PDF of the random variable Z = X + Y .
J2 . Suppose X is exponentially dis
5: INTEGRATION
STEVEN HEILMAN
1. Introduction
The integrals which we have obtained are not only general expressions which
satisfy the differential equation, they represent in the most distinct manner the
natural effect which is the object of the phenomeno
4: OPTIMIZATION
STEVEN HEILMAN
1. Optimization and Derivatives
Nothing takes place in the world whose meaning is not that of some maximum
or minimum.
Leonhard Euler
At this stage, Eulers statement may seem to exaggerate, but perhaps the Exercises in
Secti
Math 114C, Winter 2017, Solutions to HW #5
x4A.1. Prove that the partial function
f (e, u) = he (u)0 ), . . . , e (u)lh(u)
1 )i
is recursive.
Solution. We compute the graph of f :
f (e, u) = w Seq(w) & lh(w) = lh(u)
& (i < lh(u)[(w)i = e (u)i )];
this r
Math 114C, Winter 2017, Solutions to HW #3
x1C.3. Consider the definitions
g(x, y, z) = if (x = 0) then y else z,
f1 (t) = g(t, h(t), t),
f2 (t) = if (t = 0) then h(t) else t
where h(t) is some partial function. Is the equation
f1 (t) = f2 (t)
true for ev
Math 114C, Winter 2017, Solutions to HW #4
x3B.1. Assume Lemma 3B.2 and prove the Normal Form Theorem 3B.1.
Solution. (1) One direction,
for all x, f (~x) = U (yTn (e, ~x, y) = f is recursive
follows directly from the closure properties of R, specifically
Math 114C, Winter 2017, Solutions to HW #1
x1A.3. Prove that every non-empty set of natural numbers X N has a
least element.
Solution. It suffices to show that every set of natural numbers A which does
not have a least element is empty, and therefore, it
Mathematics 170A HW6 Due Tuesday, February 21, 2012.
Problems 24,25,26,31(use conditioning, not problem 27, which I will
discuss in class 2/15),32 on pages 124-133, and problem 17 on page
249.
H1 . Suppose n balls are distributed at random into r boxes. L
Problem Set 3
Due Friday, April 25.
Mathematical Logic
Math 114L, Spring Quarter 2008
1. (30 pt.) Parts (a) and (b) of Exercise 10 in Section 1.2 of the textbook.
Then do:
(c) Let
all equivalent independent subsets of
, , be wffs. Determine
, , .
2. (
HW 10
Please give complete, well written solutions to the following exercises.
For the problems from the book (denoted by #), please make sure
you solve the right problems by looking up the problem numbers
on your EBook that comes with your web assign. Th
Problem Set 8
Solutions
Mathematical Logic
Math 114L, Spring Quarter 2008
1. No, v2 is not substitutable for v0 , since v2 is a quantified variable occurring
in the term (namely, v2 ) to be substituted.
2. In both parts show by induction on simultaneously
Math 114C, Winter 2017, Solutions to HW #7
x4F.1. Classify in the arithmetical hierarchy the set
A = cfw_e | We cfw_0, 1.
01 -complete,
Solution. A is
so in the class 01 \ 01 . Proof:
x A (y)[y We = y 1],
01 .
so A is
To show the 01 -completeness, we defi
Mathematics 114C, Winter 2017
Yiannis N. Moschovakis
Second Midterm, March 7, 2017
Name (last name first):
Signature:
There are 10 problems or problem parts worth 12 point each for a total of 120 points,
20 more than the 100 points which will count as a p
Problem Set 4
Due Friday, May 2.
Mathematical Logic
Math 114L, Spring Quarter 2008
1. (20 pt.) Exercise 4 in Section 1.7 of the textbook.
2. (10 pt.) Compute
free (v1 v1 (P v1 v2 P v2 v3 ) .
3. (20 pt.) Exercise 1 in Section 2.1 of the textbook. (Include
Math 114C, Winter 2017, Solutions to HW #2
x1B.1. Prove that f : Nk N is primitive recursive if and only if f = fn
for some primitive recursive derivation (f0 , f1 , . . . , fn ).
Solution. First we show that for every primitive recursive derivation f0 ,
Problem Set 4
Solutions
Mathematical Logic
Math 114L, Spring Quarter 2008
1. The countries are C1 , C2 , . . . We can use A1 to say that C1 is red, A2 to
say that C1 is green, A3 to say that C1 is blue, and A4 to say that C1 is
yellow. And then we can use
Problem Set 2
Solutions
Mathematical Logic
Math 114L, Spring Quarter 2008
1. Using obvious shorthand notation for repeated and :
Vm Wm
(a) i=1 j=1 aij
Wm Wm
(b) i=1 j=1 aij
2. Let v be a truth assignment for a set S of sentence symbols. To show
uniqueness
HW 9
Please give complete, well written solutions to the following exercises.
For the problems from the book (denoted by #), please make sure
you solve the right problems by looking up the problem numbers
on your EBook that comes with your web assign. The
Problem Set 2
Due Friday, April 18.
Mathematical Logic
Math 114L, Spring Quarter 2008
1. (10 pt.) Let M be an m n-matrix whose entries are real numbers. Let
the sentence symbol aij represent the statement the entry in the i-th row
and j-th column is posit
Problem Set 1
Due Friday, April 11.
Mathematical Logic
Math 114L, Spring Quarter 2008
1. (30 pt.) Exercise 2 on p. 19 of the textbook.
2. (30 pt.) Exercise 3 on p. 19 of the textbook.
3. (20 pt.) Read Theorem 0B and its proof on p. 6 of the textbook. Use
Problem Set 1
Solutions
Mathematical Logic
Math 114L, Spring Quarter 2008
1. (30 pt.) By using the Induction Principle for wffs we show that every wff
has length 1, 4, 5, or length > 7. This clearly holds for sentence symbols
(they have length 1). Suppose
Mathematics 170A HW4 Due Tuesday, February 7, 2012.
Problems 55,59 on pages 68-69 and 2,4,6 on pages 119-120.
F1 . Let n be a positive integer, and let p(k) = c2k for k = 1, 2, ., n,
and p(k) = 0 otherwise. Find the value of c that makes p into a PMF.
F2
Problem Set 3
Solutions
Mathematical Logic
Math 114L, Spring Quarter 2008
1. (a) We proceed by induction on n to show that given a set consisting
of n wffs there exists an independent equivalent subset 0 of . If
n = 0, then there is nothing to show, since
Problem Set 6
Due Friday, May 16.
Mathematical Logic
Math 114L, Spring Quarter 2008
1. (20 pt.) Exercise 17 (a) in Section 2.2 of the textbook.
2. (20 pt.) Write down sentences (in the language introduced in Problem
Set 4, Exercise 7) that express the axi
Mathematics 170A HW9 Due Tuesday, March 13, 2012.
Problems 11,15,16 on pages 188-190.
K1 . Let X be uniform on [0, 1] and Y = 4X(1 X). Find the CDF
and PDF of Y .
K2 . Let X be uniform on [0, 1] and Y = log X. What is the
distribution of Y ?
K3 . The weig
Problem Set 8
Due Friday, June 6.
Mathematical Logic
Math 114L, Spring Quarter 2008
1. (10 pt.) Suppose f is a 2-place function symbol. Is v2 substitutable for v0
v
in v2 f v2 v2 = v0 ? If so, what is (v2 f v2 v2 = v0 )v20 ?
2. (10 pt.) Let x be a variabl