I
Celebration the First
Chabot College, Math 3 -
.&~
Name
[50 pts]
-~'ur~-~D-me-:-d~-t-3-I-2-0-1/
Fouquet
READ TH IS: Show your work: A correct answer with no solution is worth less than a good setup with a derailed solution, This test
is closed-book, clo
Math 8
Spring 2012
Logic vs. English
One of the goals of logic is to provide a more precise language for expressing statements that are sometimes ambiguous
in English. For example, if a restaurant menu says Your entre comes with soup or salad it probably
Math 8
Spring 2012
Nested Quantifiers as a Game
Here's a nice way to think about nested quantifiers. It's a game where, for each turn, you are trying to make the
statement true and your opponent is trying to make it false. Whoever wins determines whether
Math 8
Spring 2012
Traugott
How to Negate Quantifiers
Let S[x] be some statement containing free variable x.
1. xS[x] means that for every x in the domain S[x] is true
2. xS[x] means that for some x in the domain S[x] is true
3. xS[x] means it's not the c
Math 8
Practice Quiz 6
1. 10110101012 = _725_10
2. 34510 = _101011001_2
3. A number requires 10 bits to be represented in base 2. How many hexadecimal digits are required when the
number is represented in base 16 ? _
4. What is the base-16 representation
1. In binary search, the first recursive call is will search the list between indexes
_ and mid-1 .
first
2. Here is a recursive definition of gcd. Fill in the missing part.
gcd(n, 0) = m
gcd(n, m) = gcd(m , _)
for m > 0
nmodm
n%m
n%m
nmodm
n%m
nmodm
nmod
If f is a function from A to B and for every element b in B there is an element a in A such
that
f(a) = b
then f is onto. True
1.
2. To prove that the set of real numbers is uncountable we used the _cantor_
diagonalization argument.
3. The complement of t
Remember, in class we counted the
number of swaps for bubble sort, in the worst
case where every comparison resulted in a
swap. What was the total number of swaps for
an array of size n?
1.
Answer - n(n-1) / 2
We are attempting to prove that 3n2 + 2n
+ 10
An Example
We define f to be in O(g) iff there exist positive integers C and k such that f(n) C(g(n) whenever n > k.
So given a particular f(n) and g(n), the trick is to find suitable values for C and k ("witnesses") such that the inequality
is true.
For
Math 8
Logic and Proofs
Argument: A list of statements that are used to support a conclusion.
Example: If it just rained then the ground will be wet. It just rained. Therefore the ground is wet.
Example: If it just rained then the ground will be wet. The
Celebration
[page 2 of 4J-
4.
the First -
Math 3 -
[4 pts] (Quadric surfaces.)
Fouguet
Give the type of surface, and axis of symmetry (where applicable):
(llf(JAeJj
X-ay"t"S
flft.45lJL.OfD/
a)
2x=3y2+4z2
b)
9x2_2y2+Z2=O
c)
~LLlr5010
cePl1-er<>J fjl (0/ li
Celebration the First -
Math 3 -
[oage 3 of 41
Fouguet-
[4 pts] Find the distance traveled (arc length) of a particle moving along the curve curve
e(t) = (2e t, e -I, 2t) ,for 0 s t s 2 .
(Hint. Perfect square under the radical?):-:-:-_
U
-=-"/
tlcfw_-t)~
[page 4 of 41-
Celebration
the First -
Math 3 -
Fouguet
i
12.
[4 pts] Is the function j(x,y)
=
cfw_
x2y+
x2 +0
y2
if (x y)"* (0 0)
.
'
'contmuous
at (0,0) ?
if (x, y) = (0,0)
[i.e., present enough of an argument to answer either "Definitely No" or "Probab
Math 8, Sp. 2013
Name_
Homework 3 Due 2/27 in class
Only fill out the part below if you collaborated with someone.
Full Name(s) of person(s) you collaborated with: _
Exact time(s) and Date(s) you worked together: _
Exact location(s) where you worked: _
Yo
Math 8
Spring 2013
Existence Proofs: Constructive vs. Nonconstructive
Definition: Any Statement of the form xP(x) requires an Existence Proof. We need to show that there exists
some x that satisfies the given predicate P.
In practice there are two types o