CSE 2315 - Discrete Structures Homework 3: Sets and Combinatorics
CSE 2315 - Discrete Structures
Homework 3- Fall 2010 Due Date: Oct. 28 2010, 3:30 pm
Sets
1. Rewrite the following sets as a list of e
CSE 2315
5/2
Shortest Path and Minimal Spanning Tree
Assume we have simple weighted connected graph, where the weights
are positive. Then a path exists between the nodes x and y
How do we find a path
10/4/2014
Set symbols of set theory (,U,cfw_,.)
Symbol
Meaning /
definition
Symbol Name
Example
set
a collection of elements
A = cfw_3,7,9,14,
B = cfw_9,14,28
such that
so that
A = cfw_x | x ,x<0
AB
i
R Environment
Setup and Basics
Where to get it
Main project site is http:/www.r-project.org/
Downloading is easy:
http:/cran.revolutionanalytics.com/
R-project site contains manuals
RStudio is an
Propositional Logic, Predicates and Quantifiers
CSE-2315: Discrete Structures
Ashis Kumer Biswas
Department of Computer Science and Engineering,
University of Texas at Arlington.
CSE-2315
Discrete Str
Symbolic Representation, Propositional Logic
CSE-2315: Discrete Structures
Ashis Kumer Biswas
Department of Computer Science and Engineering,
University of Texas at Arlington.
CSE-2315
Discrete Struct
Proof by Induction
and Introduction to Recursive Definitions
CSE-2315: Discrete Structures
Ashis Kumer Biswas, Ph.D.
Department of Computer Science and Engineering,
University of Texas at Arlington.
C
Predicate Logic
CSE-2315: Discrete Structures
Ashis Kumer Biswas, Ph.D.
Department of Computer Science and Engineering,
University of Texas at Arlington.
CSE-2315
Discrete Structures
February 2, 2017
Reviewing Propositional Logic,
Introduction to Predicates, Quantifiers and
Predicate Logic
CSE-2315: Discrete Structures
Ashis Kumer Biswas
Department of Computer Science and Engineering,
University o
Lecture
21.004
.
Discrete Structures: Permutations and Combinations
Ashis Kumer Biswas, Ph.D.
Department of Computer Science and Engineering,
University of Texas at Arlington.
CSE-2315
Discrete Struct
List of Equivalence rules
Expression
Equivalent to
Name, abbreviation
P Q
P Q
QP
QP
commutative, comm
commutative, comm
(P Q) R
(P Q) R
P (Q R)
P (Q R)
associative, assoc
associative, assoc
P (Q R)
P
luture22#
Permutations, Combinations and Probabilities
Ashis Kumer Biswas, Ph.D.
Department of Computer Science and Engineering,
University of Texas at Arlington.
CSE-2315
Discrete Structures
April 10
CSE 2315
4/11
Functions
Let S and T be sets. A function (mapping) f from S to T, f: S > , T is a
subset of S x T where each member of S appears exactly as the first
compnentint of order pair. S is the
CSE 2315
3/9
Pigeonhole Principle
If more than k items are placed k bins, then at least one bin has more than one item
44
Permutations and Combinations
A = cfw_1,2,3
(1,2,3),(1,3,2),(2,1,3),(2,3,),(3,
;CSE 2315
Predicate P(x)
Quantifier(
Predicate P(x)
Quantifier( , )
Tips for translating to repdicate wff
txtbook-> page 45
Look for keywords
For all, for every, for any, for each: universal
Predicate
Francis Spears
CSE 2315-003
Inference Rules
Ax(P(x) P(t) ui
ExP(x) P(a) ei M
All flowers are plants. Sunflower is a flower. Therefore, sunflower is a plant.
P(x) is x is a flower
A is a constant symbo
CSE 2315 2/1
Hypothesis -> Conclusion
Need proof sequence to prove valid argument
Russia was a superior power, and iether France was not strong or Napoleon made an error.
Napolean did not make an erro
CSE 2315 2/15
Proof Techniques
Exhaustive Proof- Deomonstrate P implies Q for all cases
Direct Proof- Standard
Proof by Contraposition
Proof by Contradiction
Serendipity
Induction:
Principles of Mathe
CSE 2315
3/21
Permutations and Combination
Combination Order noes not matter
Permutation = (1,2,3),(1,3,2),(2,1,3).
Combination =(1,2,3)
C(n,r) or
n
Cr
C(n,r) * r! = P(n,r) C(n,r) =
P(n ,r)
r!
For n
CSE 2315
3/7/15
A = cfw_1,2,3,5,10
B= cfw_2,4,7,8,9
C= cfw_5,8,10
AuB= cfw_1,2,3,4,5,7,8,9,10
AC= cfw_1,2,3
B'
A
a b
a a
(A
C
C)= cfw_1 ,3 ,5 ,1 0
B
= cfw_ or
=
Cartesian Product
If A and B are subse
CSE 2315 1/27
Statements are sometimes called propositions
the wffs are also called propositional wffs, because the wffs represent statements
Definition of Argument
An argument is a sequence of state
CSE 2315
2/10
Proof Techniques
Informal proof methods:
Inductive reasoning
Deductive reasoning
Proof by exhaustion
Direct proof
Proof by contraposition
Proof by contradiction
Serendipity
A few
CSE 2315
1/25
1-1 Continued
A: You did not do your homework
A: You do your homework
B: You will fail
A V B You do your homework or you fail
A B If you do not do your homework, then you will fail
A B =