Consider the following sets where the domain is the set of positive integers less than equal to 100. Answer the questions.

A={x | x is a prime number}= | is a prime number

B={x | x is a even number}= | is a even number

C={x | x is a power of 2}= | is a power of 2

Find a truth set of

BC

.

Find a truth set of

AC

.

Find

ABC-

.

Find

|BC|

.

Consider

(AB)(BA)=(AB)(AB)=()

. Is it true?

P((AB)(AC))

What is

|P((CA)(CB))|

?

[5 pts] Consider two matrices

A=[3172]=3712

and

B=[2173]=2713

and answer the following questions.

Find

A+B+

.

Find

AB

.

Consider

BA

. Is it equal to

AB

?

Find the inverse matrix of

A

?

[4 pts] Consider

Z=011001100=001100110

and find

n

which makes

Z[n] []

consists of all positive integers.

Z[2][]

Z[3]

Find

n

which makes

Z[n] []

consists of all positive integers

[4 pts] Consider a function

f:ZZ:

and

f(x)=x.=.

Answer the following questions.

Is

f(x)

injective? Justify your answer.

Is

f(x)

surjective? Justify your answer.

Is there an inverse function of

f(x)

? Justify your answer.

[6 pts] Determine whether each of these functions is

O(x2)(2)

.

f(x)=152x+11=152+11

f(x)=3x2+13126=32+13126

f(x)=6x43/=643

f(x)=2x=2

f(x)=xx=

[4 pts] Consider the following sets and answer the following questions.

{1,2,3,4,, 97,98,99,100},,,,, ,,,

{1,3,5,7,,97,99},,,,,,

{2,4,6,8,,96,98,100}{,,,,,,,}

[BONUS QUESTION: 3 pts] Consider two sorting algorithms and a sequence

A

below Sort the sequence

A

using insertion sorting and show every change in

A

.

A={24,37,11,21,99,52,53,74}={24,37,11,21,99,52,53,74}

[ALGORITH: INSERTION-SORT(

A

)]

1 for

j = 2 =

to

A.length.

2

key = A[j] = []

3 // insert

A[j][]

into the sorted sequence

A[1 j1][ ]

4

i = j1 =

5 while

i > 0 >

and

A[i] > key[] >

6

A[i+1]= A[i][+]= []

7

i=i1=

8

A[i+1]=key[+]=