Celebration the First
Chabot College, Math 3 -
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
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
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
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
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 .
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
If f is a function from A to B and for every element b in B there is an element a in A such
f(a) = b
then f is onto. True
2. To prove that the set of real numbers is uncountable we used the _cantor_
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?
Answer - n(n-1) / 2
We are attempting to prove that 3n2 + 2n
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
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
[page 2 of 4J-
the First -
Math 3 -
[4 pts] (Quadric surfaces.)
Give the type of surface, and axis of symmetry (where applicable):
cePl1-er<>J fjl (0/ li
Celebration the First -
Math 3 -
[oage 3 of 41
[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?):-:-:-_
[page 4 of 41-
the First -
Math 3 -
[4 pts] Is the function j(x,y)
if (x y)"* (0 0)
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
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: _
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