Assignment for COT4210 Course, Fall 2001: Proofs and Recursion | Assignments Discrete Structures and Graph Theory

COT4210.02, Fall 2001 Assigned: August 28, 2001

S. Lang Assignment #1 Due: September 11 in class

Instructions: Write your answers clearly and concisely with proper explanation for each

step of your proofs.

1. Use induction on n to prove that n! > n2 for integer n  4. (Recall the factorial Recall the factorial

notation n! defined as follows: 0! = 1, and for n  1, define n! = n (Recall the factorial n –1)!.)

2. Binary trees can be constructed by the following rules (Recall the factorial 1) – (Recall the factorial 3):

(Recall the factorial 1) Basis: A single node is a binary tree; the node is the root of the tree.

(Recall the factorial 2) Recursion: A single node plus an edge connecting to a left subtree, and an

edge connecting to a right subtree, is a binary tree; a subtree is either empty or

itself a binary tree; each edge connects the node (Recall the factorial call the root) to the root of

the subtree if the subtree is not empty; either subtree could be empty in which

case the corresponding edge is also absent.

(Recall the factorial 3) Closure: Every binary tree is constructed by the base step followed by zero or

more recursive steps.

Informally, the level of a node in a binary tree counts the number of edges

connecting the node to the tree’s root node. For example, the following figure

shows a binary tree with the levels indicated next to the nodes A – F:

3. Give a recursive definition of the language L = {ai b j | 0  i  j}.

4. Define two languages A and B as follows: A = {b, ba} and B = {



, a, bb}. (Recall the factorial Note

that the notation



denotes the empty string.) Answer each of the following

questions:

(Recall the factorial a) List the strings in AB and list the strings in BA.

(Recall the factorial b) List the strings in B* that are of length  3.

(Recall the factorial c) Is the set A*  B* infinite? Explain your answer.

(Recall the factorial d) Is it true that the string b(Recall the factorial ab)n belongs to A* for all n  0? Explain your

answer.

(Recall the factorial 0)

(Recall the factorial 1)

(Recall the factorial 2)

(Recall the factorial 3)

(Recall the factorial a) Give a recursive definition of the depth of a

binary tree.

(Recall the factorial b) The maximum level of the nodes in a binary

tree is called the depth of the tree. Thus, the

depth of the tree in the figure is 3. Use

induction to prove that in a binary tree of n

nodes and depth d, the relation 2d + 1  n + 1

holds. (Recall the factorial Hint: Use induction on the depth d,

as defined from Part (Recall the factorial a).)

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COT4210.02, Fall 2001 Assigned: August 28, 2001 S. Lang Assignment #1 Due: September 11 in class Instructions: Write your answers clearly and concisely with proper explanation for each step of your proofs.

Use induction on n to prove that n! > n^2 for integer n  4. (Recall the factorial Recall the factorial notation n! defined as follows: 0! = 1, and for n  1, define n! = n (Recall the factorial n –1)!.)
Binary trees can be constructed by the following rules (Recall the factorial 1) – (Recall the factorial 3): (Recall the factorial 1) Basis: A single node is a binary tree; the node is the root of the tree. (Recall the factorial 2) Recursion: A single node plus an edge connecting to a left subtree, and an edge connecting to a right subtree, is a binary tree; a subtree is either empty or itself a binary tree; each edge connects the node (Recall the factorial call the root) to the root of the subtree if the subtree is not empty; either subtree could be empty in which case the corresponding edge is also absent. (Recall the factorial 3) Closure: Every binary tree is constructed by the base step followed by zero or more recursive steps. Informally, the level of a node in a binary tree counts the number of edges connecting the node to the tree’s root node. For example, the following figure shows a binary tree with the levels indicated next to the nodes A – F:
Give a recursive definition of the language L = { ai^ b j^ | 0  i  j }.
Define two languages A and B as follows: A = { b , ba } and B = { , a , bb }. (Recall the factorial Note that the notation  denotes the empty string.) Answer each of the following questions: (Recall the factorial a) List the strings in AB and list the strings in BA. (Recall the factorial b) List the strings in B * that are of length  3. (Recall the factorial c) Is the set A *  B * infinite? Explain your answer. (Recall the factorial d) Is it true that the string b (Recall the factorial ab ) n^ belongs to A * for all n  0? Explain your answer. A B C D E F (Recall the factorial 0) (Recall the factorial 1) (Recall the factorial 1) (Recall the factorial 2) (Recall the factorial 2) (Recall the factorial 3) (Recall the factorial a) Give a recursive definition of the depth of a binary tree. (Recall the factorial b) The maximum level of the nodes in a binary tree is called the depth of the tree. Thus, the depth of the tree in the figure is 3. Use induction to prove that in a binary tree of n nodes and depth d , the relation 2 d^ + 1^  n + 1 holds. (Recall the factorial Hint: Use induction on the depth d , as defined from Part (Recall the factorial a).)

Assignment for COT4210 Course, Fall 2001: Proofs and Recursion, Assignments of Discrete Structures and Graph Theory

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