MATH 302. Discrete Mathematics Extra-credit Assignment 1.
Please show your argument and computation. Calculators and computers are not permitted. 1. Prove that 2 + 3 is irrational. 2. Write down the set P(P(P(P()? How many elements are there in this set?
MATH 302 Discrete Mathematics
Assignment 2.
Due on Friday, September 19, 2014
Read: Sections 3.2, 1.1 1.2.
Denition: Write down the denitions for the following terms.
proposition
truth value
converse
contrapositive
inverse
Problems to be graded:
3.2/ 41,
MATH 302 Discrete Mathematics
Assignment 6.
Due on Monday, October 27, 2014
Read: Sections 5.15.3
Denition: Write down the denitions for the following terms.
Principle of Mathematical Induction
Principle of Strong Induction
the Well-Ordering Property
recu
MATH 302 Discrete Mathematics
Assignment 5.
Due on Friday, October 17, 2014
Read: Sections 2.3, 2.4, 2.5
Denition: Write down the denitions for the following terms.
[5 points]
a function from A to B
a function is one-to-one
a function is onto
the set A an
MATH 302 Discrete Mathematics
Assignment 1.
Due on Friday, September 12, 2014
Read: Sections 3.1-3.2.
Denition: Write down the denitions for the following terms.
algorithm,
greedy algorithm,
f (x) is O(g(x),
f (x) is (g(x),
f (x) is big-Theta of g(x)
Prob
MATH 302 Discrete Mathematics
Assignment 9.
Due on Friday, November 21, 2014
Read: Sections 6.3-6.4
Denition: Write down the denitions for the following terms.
The Binomial Theorem
Pascals Identity
Vandermondes Identity
Problems to be graded:
6.3/ 12, 18,
MATH 302 Discrete Mathematics
Assignment 10.
Due on Monday, December 1, 2014
Read: Sections 6.5, 8.58.6. 2.6
Denition: Write down the denitions
for the following terms.
[5
points].
The number of r-combinations from a set with n elements when
repetition of
MATH 302 Discrete Mathematics
Assignment 8.
Due on Friday, November 14, 2014
Read: Sections 8.2, 8.3, 6.1,
Denition: Write down the denitions for the following terms.
[5 points]
Master Theorem (See page 532. Please be aware that the version
given in class
Mathematical Induction
Dominoes: why are all the dominoes falling down?
Because: they are listed one-by-one in close positions, and there is an initial push.
Inductive principle:
P (1)
k(P (k) P (k + 1)
nP (n)
To prove nP (n), we complete two steps:
1. Ba
Algorithms and Functions
Algorithm: a nite set of precise instructions for performing a computation or for solving a
problem.
It is the black-box with three properties:
Deniteness: each step is dened precisely
Finiteness: end after a nite many steps
Ge
Lecture on Logic and Proofs
Building blocks of logic: propositions: a is a declarative sentence that is either true or false,
but not both.
Examples: These are propositions:
1. September 9, 2009 is Wednesday.
2. September 9, 2009 is Friday.
3. Today is We
Lecture on Rules of Inference
Discuss example: what do you know from the following facts?
1. X and Y both need 87 in nal to reach 450 pts.
2. X got 92 in nal.
3. Y didnt get A.
Three basic rules
Modus Ponens (p, p q) = q.
Modus Tollens (q, p q) = q.
Hy
Lecture on Proofs
Common Mistakes fallacies
Example 1. We know that if an integer is even, then its square is even. Since n2 is even, then n
is even.
(From p q and q, it claims p: fallacy of arming the conclusion.
Example 2. If it rains, you will pick me
Functions
f : A B: a function f from A to B is an assignment of exactly one element of B to each
element of A. Write f (a) = b.
e.g. Real functions, birthday, height in centimeters, SSN.
A is the domain, B is the codomain, a is a preimage of b, and b is t
Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2015
Math 304 Problem Set 3
Issued: 09.18
Due: 09.25
3.1. Find an elementary matrix E such that EA = B for
4 2 3
4 2 3
0 2 ,
0 2 .
A= 1
B= 1
2
3 1
0
3 5
3.2. Evaluate the following
MATH 302 Discrete Mathematics
Assignment 7.
Due on Wednesday, October 28, 2015
Read:
Sections 5.2, 5.3, 8.1,
Principle of Strong Induction
the Well-Ordering Property
recursive or inductive denition
formula of sums of geometric progression (Section 2.4)
Pr
t e
r
8
I t e
i s
x
a
g
m
o
e
n
i e v
d
d
i
o
G
a
m
a
m
o
,
n
c h
a
l a
r g
e
a
c c o
o
f
l v
Ma
i v
C
i n
v
e
u
o
n
s
d
t e r m
s
f o
i t i a
l
t a
r
i n
b
x
t h
o
d
s
s
o
i n
t o
o
d
e t h
n
r e
n
o
m
a
t e d
t i o
b
a
a
s o
p
o
i n
i m
b
b
t e
l v
r o
MATH 302 Discrete Mathematics
Assignment 3.
Due on Friday, September 26, 2014
Read: Sections 1.3 1.5
Denition: Write down the denitions for the following terms.
[5 points]
the universal quantication of P (x), universal quantier
counterexample
the existent
MATH 302 Discrete Mathematics
Assignment 7.
Due on Monday, November 3, 2014
Read: Sections 8.1, 8.2
Denition: Write down the denitions for the following terms.
[5 points]
linear homogeneous recurrence relation of degree k with constant coecients
linear no
MATH 302 Discrete Mathematics Extra-credit Assignment 2.
Please show your argument and computation. Calculators and computers are not permitted.
2 1. Let fn be the n-th Fibonacci number. Prove that f0 f1 + f1 f2 + + f2n-1 f2n = f2n when n is a positive in
MATH 302 Discrete Mathematics Assignment 1. Due on Wednesday, September 8, 2010
Read: Sections 3.1-3.2 Definition: Write down the definitions for the following terms. algorithm, greedy algorithm, f (x) is O(g(x), f (x) is (g(x), f (x) is big-Theta of g(x)
MATH 302 Discrete Mathematics Assignment 2. Due on Wednesday, September 15, 2010
Read: Sections 1.1, and 1.2. Definition: Write down the definitions for the following terms. proposition truth value converse, contrapositive, and inverse tautology contradic
MATH 302 Discrete Mathematics Assignment 3. Due on Wednesday, September 22, 2010 Read: Sections 1.3 and 1.4 Definition: Write down the definitions for the following terms.
[5 points] the universal quantification of P (x), universal quantifier, counterexam
MATH 302 Discrete Mathematics Assignment 4. Due on September 29, 2010 Read: Sections 1.51.7. Definition: Write down the definitions for the following terms.
fallacy of affirming the conclusion fallacy of denying the hypothesis even and odd for integers ra
MATH 302 Discrete Mathematics Assignment 5. Due on Wednesday, October 6, 2010 Read: Sections 2.12.3 Definition: Write down the definitions for the following terms.
[5 points]
the difference of A and B, where A and B are sets a function from A to B a funct
MATH 302 Discrete Mathematics Assignment 6. Due on Wednesday, October 20, 2010 Read: Sections 2.4, 4.1, 4.2 Definition: Write down the definitions for the following terms.
the sets A and B have the same cardinality countable Principle of Mathematical Indu
MATH 302 Discrete Mathematics Assignment 7. Due on Wednesday, October 27, 2010 Read:
Sections 4.3, 7.1-7.2 For nonhomogeneous recurrence relations, you are only required to know how to solve the first order ones.
Definition: Write down the definitions for
MATH 302 Discrete Mathematics Assignment 8. Due on Wednesday, November 3, 2010. Read: Sections 7.3, 5.1, 5.3. Definition: Write down the definitions for the following terms.
The product rule The sum rule r-permutation r-combination
[5 points]
Problems to
MATH 302 Discrete Mathematics Assignment 11. Due on Wednesday, November 24, 2010 Read: Sections 8.48.6, (up to the end of page 567), Definition: Write down the definitions for the following terms on relations (Chapter 8). [5 points] composite of (two rela
MATH 302 Discrete Mathematics Assignment 12. Read: Sections 8.48.6, (up to the end of page 567), section 5.2 Denition: Write down the denitions for the following terms on relations (Chapter 8). [5 points] composite of (two relations) n-ary relation and it