CS 314

FORMAL LANGUAGES AND AUTOMATE

THEORY

Introduction

1

TA. Anfal M. AlHazzaa

Grading ( 10 points)

Quizzes (7 points)

Participation ( 2 point)

Attendance ( 1 point )

Quiz2 Quiz1

3.5 3.5

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CS 314

FORMAL LANGUAGES AND AUTOMATE

THEORY

Sheet # 1 Exercises (From the textbook –Introduction to The

Theory of Computation – Ch. 0)

4

TA. Anfal M. AlHazzaa

Question 1

Examine the following formal descriptions of sets so

that you understand which members they contain.

Write a short informal English description of each set.

a-{1,3,5,7, ... }

odd natural numbers

b. {..., -4, -2,0,2,4,...}

even integers numbers

Question 1 cont.

c. {n |n = 2m for some m in N}

even numbers

e. {w | w is a string of 0's and 1's and w equals

the reverse of w}

e.x 0110

f. {n| n is an integer and n = n + 1}

Z

Question 2

Write formal descriptions of the following sets

a. The set containing the numbers 1, 10, and 100

{n|n = 𝟏𝟎𝒎 for m є {0,1,2}} or

{0,1} {i| i ={0,1}}

b. The set containing all integers that are greater than 5

{n|n > 5 for n є Z }

c. The set containing all natural numbers that are less than 5

{n|n < 5 for n є N }

Question 2 cont.

d. The set containing the string aba

{a,b}

e. The set containing the empty string

{ } or

f. The set containing nothing at all

Ø

FYI 1

It says "the set of all x's, such that x is greater than 0".

In other words any value greater than 0

Notes:

The "x" is just a place-holder, it could be anything, such as { q | q > 0 }

Some people use ":" instead of "|", so they would write { x : x > 0 }

FYI 2

Question 3

Let A be the set {x, y, z} and B be the set {x, y}.

a. Is A a subset of B?

No it is not

b. Is B a subset of A?

Yes it is

c. What is A ∩ B?

{x,y}

Question 3 cont.

d. What is A U B?

{x,y,z}

e. What is A × B?

{(x,x),(x,y),(y,x),(y,y),(z,x),(z,y)}

f. What is the power set of B

{Ø,{x},{y},{x,y}}

Question 4

Let x be the set {1, 2, 3, 4, 5} and y be the set {6, 7, 8, 9, 10}. The unary function

f: x y and the binary function g: X ×Y Y are described in the following tables.

n f(n) g 6 7 8 9 10

1 6 1 10 10 10 10 10

2 7 2 7 8 9 10 6

3 6 3 7 7 8 8 9

4 7 4 9 8 7 6 10

5 6 5 6 6 6 6 6

a. What is the value of f (2)?

7

Question 4 cont.

b. What are the range and domain of f?

the range {6,7} the domain {1,2,3,4,5}

c. What is the value of g(2, 10)?

6

d. What are the range and domain of g?

the range {6,7,8,9,10}

the domain {(1,6),(1,7),(1,8),(1,9),(1,10),(2,6)(2,7),(2,8),(2,9),(2,10),(3,6),(3,7),(3,8),(3,9),(3,10),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10)}

e. What is the value of g(4, f(4))?

8

Question 5

Write a formal description of the following graph

G = (V, E)

any order

G={{1,2,3,4,5,6},{(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(

3,4),(3,5),(3,6)}

Question 5 cont.

a. what is the degree of node 1?

3

Question 6

let x = 01101; y = 110 two strings over the set ∑= {0,1},

a.what is xy ? xy=01101110

b. what is the length of the string x ?

|01101|= 5

c. what is | | ? | | = 0

d. what is the reverse of y ? 011

Question 7

𝞢1 = {a, . . . , z} and 𝞢2 = {0, . . . , 9} are alphabets ,

choose the correct answers :

a. abb is string over :

1- 𝞢1

2- 𝞢𝟐

3- none from the above

4- both 1 and 2

Question 7 cont.

123 is string over :

1- 𝞢1

2- 𝞢𝟐

3- none from the above

4- both 1 and 2

Question 7 cont.

is string over :

1- 𝞢1

2- 𝞢𝟐

3- none from the above

4- both 1 and 2