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
Discrete Structures
CSE 2315
Lecture 16
Practice with Sets
Dr. Dimitrios Zikos
Department of Computer Science and Engineering
Set Fundamentals
Representing Sets
The extensional method - Explicitly enumerate all the elements of the set;
e.g., A = cfw_1,
Discrete Structures
CSE 2315
Lecture 12 Mathematical Induction
Dimitrios Zikos
Department of Computer Science and Engineering
Introduction to induction
Informal proof techniques permit the implicit use of general
domain knowledge to simplify the proof of
Discrete Structures
CSE 2315
Lecture 10 Proof
Dimitrios Zikos
Department of Computer Science and Engineering
Proof
You already know that arguments are statements having the form
P Q or more generally P(x) Q(x).
Formal logic allows to perform detailed, d
Discrete Structures
CSE 2315
Lecture 17 Counting
Dimitrios Zikos
Department of Computer Science and Engineering
(A U ) (A S)=
Prove that this is an empty set
Dimitrios Zikos
Discrete Structures
2
Prove that this is an empty set
[A (B U )] (S U B)
Dimitrio
Discrete Structures
CSE 2315
Lecture 13 Recursive and
Introduction to Sequences
Dimitrios Zikos
Department of Computer Science and Engineering
Recursion
Dimitrios Zikos
Discrete Structures
2
What is lateral thinking ?
Lateral thinking is about concepts t
Discrete Structures
CSE 2315
Lecture 15
Basic Theory of Sets (and some more
examples on recursion)
Dimitrios Zikos
Department of Computer Science and Engineering
Some more examples on recursion
2
4
12
48
240
F(1)=2
F(2)=4
= 2*2
F(3)=12
=3*4
F(4)=48
=4*12
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
Discrete Structures
CSE 2315
Relations
Dimitrios Zikos
Department of Computer Science and Engineering
Binary Relations
Definition
A binary relation R between the sets A and B, is a subset of their Cartesian
product A x B. We say that a A is related to b
Discrete Structures
CSE 2315
Lecture 24 Functions
Dimitrios Zikos
Department of Computer Science and Engineering
What is an order?
We dene an order as a transitive relation R that can
be either
1. reexive and antisymmetric (a weak partial order)
, written
Discrete Structures-Homework 3
First Name: Mohammed
Last Name: Ali
ID: 1001241690
Please answer all questions and submit via the Blackboard before April 10th, 2016
1.
Prove by ordinary induction the following: For all n (n > 1), 8n 3n is divisible by 5.
2
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Homework 3 Solutions
1. Prove by ordinary induction the following: For all n (n > 1), 8n 3n is divisible by 5.
Base case: n>1 n= 2: P(2) = 8*2 -3*2 = 16 6 = 10 = 5*2
proved
Induction step: n = k P(k) = 8k 3k = 5m, we assume it is correct
Check for n = k +
Discrete Structures Spring Semester 2016
HOMEWORK #4
Name:
ID:
Please submit via the Blackboard, by Wednesday, May 4th
PROBLEM 1
Five cities A, B, C, D and F are connected as shown below. Train Ace provides fast trips, connecting neighboring
cities only.
Functions (ii)
Discrete Structures
Dimitrios Zikos
Functions
Discrete Structures
Composition of Functions
"Function Composition" is applying one function to the results of
another
The result of f() is sent through g()
(g f)(x) = g(f(x)
Example of composit
Homework 2 Solutions
1. Prove the validity of the following arguments
I. x R(x) [x R(x) (x)S(x)] x S(x)
Proof:
1. ()
2. [() ()]
3.()
4. () ()
5. ()
6. ()
hyp
hyp
1, ui
2, ui
3,4, mp
5, ug
II. x [P(x) V Q(x)] [(x) P(x)] x Q(x)
Proof:
1. [()()]
hyp
2. [()]
Discrete Structures
CSE 2315 (Fall 2014)
Lecture 19 Permutation and
Combination
DIMITRIOS ZIKOS
DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING
Discussing permutation a bit more
This problem consists of a sequence of events and
a solution thus involves the
Discrete Structures
CSE 2315 (Fall 2014)
Lecture 11
-Practice with Proof Techniques
Dimitrios Zikos
Department of Computer Science and Engineering
Exhaustive proof.
Conjecture: All odd numbers between 3 and 25 are either prime
numbers or the product of ex
Discrete Structures
CSE 2315
Lecture 13 Recursive and
Introduction to Sequences
Dimitrios Zikos
Department of Computer Science and Engineering
Recursion
Dimitrios Zikos
Discrete Structures
2
What is lateral thinking ?
Lateral thinking is about concepts t
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 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 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
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 - 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
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 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
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
Discrete Structures
CSE 2315
Lecture 18 Counting
Dimitrios Zikos
Department of Computer Science and Engineering
Decision Trees
The multiplication principle cannot be used if the number of
choices at a given stage depends upon the exact choice made
in the
Introduction to Binary
Algebra
Dsicrete Structures
CSE-2315
Generalization of propositional logic. A minor generalization of
propositional logic.
In general, algebra is any mathematical structure satisfying is any
mathematical structure satisfying certa
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