Logic
Statements, negations, connectives, truth tables, equivalent statements, De Morgans
Laws, arguments, Euler diagrams
Part 1: Statements, Negations, and Quantified Statements
A statement is a sentence that is either true or false but not both simultan
MATH 302 Discrete Mathematics
Solution 4. Sections 1.6-1.7
Problems to be graded:
1.6/ 14, 20, 24, 26, 27, 35
1.7/ 9, 12, 17, 18
Solutions.
Section 1.6/14. (Answers are not unique, depending on how the logical form is expressed. )
(a) Universal instantia
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
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
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
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
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 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 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 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
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
MATH 302 Discrete Mathematics
Solution 1. Sections 3.1 3.2.
1. 3.1/8 Start with k := 0, and L := . For i = 1 to n, if ai is even and ai > L, then
let L = ai and k = i. Output k.
2. 3.1/9 Start with AN S = 1. For i = 1 to n, check whether ai = an+1i . If y
MATH 302 Discrete Mathematics
Solution 5. Sections 2.12.3
Problems to be graded:
2.1/ 10, 26,
2.2/ 17(a), 30.
2.3/ 14, 22, 34, 35, 38, 44.
Solutions.
Section 2.1/10. T, T, F, T, T, T, F.
Section 2.1/26. For any element (a, b) A B, by definition we know
MATH 302 Discrete Mathematics
Solution 2. Sections 3.2, 1.1 1.2.
1. 3.2/41 (This is a hard problem. The student must use both the conditions that f
is O(g) and that both f and g are increasing and unbounded. )
Proof. Since f is O(g), we know that there ar
MATH 302 Discrete Mathematics
Solution 3. Sections 1.3 1.5.
1. 1.3/14 No. You can use the truth table, or just show that when p is false and q is
true, the proposition p (p q) is true while q is false, hence the whole proposition
is false. Not a tautology
MATH 302 Discrete Mathematics
Assignment 3.
Due on Wednesday, September 23, 2015
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 exist
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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
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
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
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 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 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 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 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 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 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 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 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