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 elements. a) cfw_x|(y )(y N x = y 3 x < 30) b) cfw_x|x i
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 with minimum weight?
For example, cities connected by r
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
intersection
objects that belong to
set A and set B
A B
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 independent IDE for R
You need to have R installed to
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 Structures
January 26, 2017
1 / 16
Propositional Logic
Goa
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 Structures
January 23, 2017
1 / 18
January 23, 2017
2 / 18
St
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.
CSE-2315
Discrete Structures
February 28, 2017
1 / 13
Pr
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
1 / 13
Predicates
Predicate says a property of a subjec
Reviewing Propositional Logic,
Introduction to Predicates, Quantifiers and
Predicate Logic
CSE-2315: Discrete Structures
Ashis Kumer Biswas
Department of Computer Science and Engineering,
University of Texas at Arlington.
CSE-2315
Discrete Structures
Janu
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 Structures
April 5, 2017
1 / 20
Outlines
1
Permutations and C
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 (Q R)
(P Q) (P R)
(P Q) (P R)
distributive, dist
distri
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, 2017
1 / 25
Permutations and Combinations
What is the
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 domain and T is codomain. If (s,t)
belongs to the func
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,),(3,1,2),(3,2,1)
(1,2,3)
Permutations
An ordered arrangemen
;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 Logic
Similar to propositional logic for solving argum
CSE 2315 - Discrete Structures Homework 1: Propositional Calculus and Predicate Logic
CSE 2315 - Discrete Structures
Homework 1- Fall 2010 Due Date: Sept. 16 2010, 3:30 pm
Statements, Truth Values, and Tautologies
1. Which of the following are statements
CSE 2315 - Discrete Structures Homework 1: Propositional Calculus and Predicate Logic
CSE 2315 - Discrete Structures
Homework 1- Solutions - Fall 2010 Due Date: Sept. 16 2010, 3:30 pm
Statements, Truth Values, and Tautologies
1. a) Is a statement. b) Is a
CSE 2315 - Discrete Structures Homework 2: Predicate Calculus and Proof Techniques
CSE 2315 - Discrete Structures
Homework 2- Fall 2010 Due Date: Oct. 7 2010, 3:30 pm
Proofs using Predicate Logic
For all your predicate logic proofs you can use only the ru
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 symbol (Sunflower)
Q(x) x is a plant
Ax[P(x) Q(x)] hyp
P(a)
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 error, but if the army did not fail, then France was string
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 Mathematical Induction
First Principle:
P(1) is true
(Ak)P(k
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 by n grid, C(2n,n)
For duplicate divide
Permutation for
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 subsets of S, then the cartesain product (cross product) of
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 statements in which the conjunction of the initial
statement
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 terms for proof:
Axioms: Statements that are assumed t
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 = A V B
Conjunction: And;But;Also;in addition;moreover A
lecture
020
section
Counting and Principles of Inclusion and Exclusions
CSE-2315: Discrete Structures
Ashis Kumer Biswas, Ph.D.
Department of Computer Science and Engineering,
University of Texas at Arlington.
CSE-2315
Discrete Structures
April 3, 2017
1