Tuesday, March 17th 2015
MATH 15A
Final Exam
Follow these instructions carefully.
1. No calculators or other electronic computational aids may be used during the exam.
2. You may have one page of notes, but no books or other assistance.
3. Write your name
1)
Let the following statements be given.
p = There is water in the cylinders.
q = The head gasket is blown.
r = The car wont start.
(a) Translate the following statement into symbols of formal logic
If the head gasket is blown and theres water in the cyl
MATH 15A
Aisenberg
Introduction to Discrete Mathematics
Recursion and Induction
James Aisenberg
[email protected]
February 10, 2015
Recursive denitions
MATH 15A
Aisenberg
A recursive denition of a set has
one or more bases cases (saying that some concre
MATH 15A
Aisenberg
Sequences
Introduction to Discrete Mathematics
Sequences and Mathematical Induction
James Aisenberg
[email protected]
February 3, 2015
Miscellaneous
MATH 15A
Aisenberg
Sequences
Injectivity and surjectivity
Proving if and only if stat
MATH 15A
J. Aisenberg
Denitions
Direct proof
Proof by
contraposition
Introduction to Discrete Mathematics
Proof Strategies I
Proof by
contradiction
(Dis)proof by
counterexample
Peer work
James Aisenberg
[email protected]
Review:
quantiers
January 20, 20
MATH 15A
Practice Exam for Midterm 1
Winter, 2016
1. LOGIC
1. Let s be the following statement.
If I see a penguin and a leopard seal then I am in antarctica.
(a) Let P be I see a penguin, L be I see a leopard seal and A be I am in Antarctica
(P L) A
(b)
MATH 15A
J. Aisenberg
More about
quantiers
Proofs in
propositional
logic
Introduction to Discrete Mathematics
Quantiers (continued), and Proofs in Propsitional Logic
James Aisenberg
[email protected]
January 15, 2015
Outline
MATH 15A
J. Aisenberg
More a
MATH 15A
J. Aisenberg
From signals
to sentences
Possible
worlds and
truth tables
Introduction to Discrete Mathematics
Logic I
James Aisenberg
[email protected]
January 8, 2015
Outline
MATH 15A
J. Aisenberg
From signals
to sentences
Possible
worlds and
t
MATH 15A
Aisenberg
Basic
Counting
Selections
and arrangements
Introduction to Discrete Mathematics
Counting with numbers
Counting
practice
James Aisenberg
[email protected]
March 3, 2015
MATH 15A
Aisenberg
Basic
Counting
Selections
and arrangements
Coun
MATH 15A
Aisenberg
Counting
with
functions
Innite sets
Introduction to Discrete Mathematics
Counting with functions
James Aisenberg
[email protected]
March 5, 2015
MATH 15A
Aisenberg
Counting
with
functions
Innite sets
Example
Make a tables showing that
MATH 15A
J. Aisenberg
Introduction to Discrete Mathematics
Logic II: Predicate logic
James Aisenberg
[email protected]
January 15, 2015
Opening question
MATH 15A
J. Aisenberg
What is the negation of the sentence All cows are green.
1
None of the cows ar
MATH 15A
Aisenberg
Algorithms
and
modular
arithmetic
Partial
orders and
total orders
Introduction to Discrete Mathematics
Algorithms, Partial Orders, Total Orders
James Aisenberg
[email protected]
February 26, 2015
MATH 15A
Aisenberg
Algorithms
and
modu
MATH 15A
Aisenberg
Introduction to Discrete Mathematics
Algorithms and correctness (contd), relations and
equivalence relations
James Aisenberg
[email protected]
February 17, 2015
MATH 15A
Aisenberg
Recall that an equivalence relation R on a set S has t
MATH 15A
Aisenberg
Algorithms
and
induction
Relations
and
equivalence
relations
Introduction to Discrete Mathematics
Algorithms and correctness (contd), relations and
equivalence relations
James Aisenberg
[email protected]
February 17, 2015
Converting t
MATH 15A
Worksheet 3
Exercise 1.
For the sequence given below, write out the rst few values of the sequence, use those
values to guess a closed-form expression and then prove that the closed-form expression
you found is correct.
The function f dened by th
MATH 15A
Practice Exam for Midterm 1
Winter, 2016
1. LOGIC
1. Let s be the following statement.
If I see a penguin and a leopard seal then I am in antarctica.
(a) Write this in terms of formal logic.
(b) Write the converse of this statement in English.
(c
MATH 15A
Practice Exam for Midterm 1 part 2
Powerset practice Winter, 2016
Powersets and inclusion versus subset.
Remember that for a set S, P(S) = cfw_A|A S.
Determine whether each of the following are true or false.
1. cfw_1, 2, 10, P(N)
2. N P(Z)
3. Z
11.1:3
11.1:5
a.
b.
c.
d
Text Chapter 1.3
Problem 8
(a) (x) (y) (N(x) P(x, y)
(b) (x) (y) (N(x) P(x, y)
(c) There is some non-zero integer x such that for all integers y, xy1
(d) The statement of (b) is true in
MATH 15A
Homework 4 solutions
Winter, 2016
1. Chapter 1.5 # 2
Give a direct proof of
Let a, b, c be integers. If a|b and a|c then a|(bc).
Proof. Let a, b, c be integers. Suppose a|b and a|c. Then, b = ak and c = a for integers k, .
Then bc = (ak)(a ) = a(
Math 15A HW 2 Solutions
January 29, 2016
Chapter 1 Section 1
#8
An integer is divisible by 10 if and only if its last digit is a zero.
#12
a
T
T
F
F
b
T
F
T
F
ab ab
T
T
T
F
T
F
F
F
(a b) (a b) (a b)
F
F
T
T
T
T
T
F
a b
F
T
T
F
Since the last two columns a
1)
Let the following statements be given.
p = There is water in the cylinders.
q = The head gasket is blown.
r = The car wont start.
(a) Translate the following statement into symbols of formal logic
If the head gasket is blown and theres water in the cyl
MATH 15A
Practice Midterm I
Follow these instructions carefully.
1. No calculators or other electronic computational aids may be used during the exam.
2. You may have one page of notes, but no books or other assistance.
3. Write your name, PID, and sectio
MATH 15A
Practice Midterm I
Follow these instructions carefully.
1. No calculators or other electronic computational aids may be used during the exam.
2. You may have one page of notes, but no books or other assistance.
3. Write your name, PID, and sectio
MATH 15A
Worksheet 4
Exercise 1.
Identify the error in the following proof of the claim that all horses are the same color.
Let P (n) be the predicate any set of n many horses all have the same
color. We will show (n)(P (n) by induction. The base case is
MATH 15A
Discrete
Mathematics
Prof. Shachar Lovett
1
2
Todays Topics:
1.
2.
3.
Graphs
Some theorems on graphs
Eulerian graphs
3
Graphs
Model
Basic
relations between pairs of objects
ingredient in many algorithms:
Network routing, GPS guidance,
Simulati
Solution Set 14 - Functions
4) B
5) B
6) B
7) A
8) B
10) D
11) B
12) C
14) E - All are injective.
15) E - B, C and D are not surjective
16) A
17) B
18) B
34) C - The function f : Z E with f (x) = 2x is injective, and the function f : E Z with f (x) = x/2
MATH 15A
Discrete
Mathematics
Prof. Shachar Lovett
1
2
Todays Topics:
1.
2.
3.
Relations
Equivalence relations
Modular arithmetics
3
1. Relations
4
Relations are graphs
Think
of relations as directed graphs
xRy means there in an edge xy
Is
A.
B.
C.
D.