## Question:

1 (a) Find a bijection between the invervals (0. 0) and (0. 1.) Use the Schr?oderBernstein to prove that the intervals (0. 0) and (0. 1) have the identical cardinality.

Q2 Examine the groups (Q xQ, +), and (Q2).

a. b Q with (a. b) + [c,] = (a+ c, + d); Q[2] = *a +?b

2

This will show that the two groups are identical.

Q3 Write Cayley tables for (Z9+) and (Z9*).

Briefly explain why / pourquoi not (you do NOT need a formal proof).

Q4 Which of the following is a 6 digit PIN?

(a), The code includes six different digits

(b) Code contains only odd numbers. Repeatable digits allowed

(c) At least one of the codes contains an odd number and at most one of an even number. Digits can be repeated

(d), The code contains three odd numbers and three even ones

(e), the code contains six different numbers and does not contain two odd numbers immediately

Following one another (e.g.: 258746 allowed, but 259406 prohibited)

You should note that parts (a),-(e), are five distinct, unrelated questions. Therefore, the rule for part a does not apply parts (b),-(e) and so on.

Q5 Let X represent an arbitrary set and R represent an arbitrary relationship on X.

A new relation Q is defined for X using the rule:

X, y, X, x Q, y only if Z X such that R z and R y.

Demonstrate it

(a) Q is an equivalence relations if R is an equivalence.

(b) R can be a partial ordering, so Q can also be a partial ordering

## Answer:

[0, ] & [ 0,1]

By f(x), = 1/2

Therefore, f does map from [0,1] to[1,1]

Also, f-1 = since then x

01 so f-1 Maps [1, ] to[0,1]

Additionally

F(f-1 [x])= f()= =

Define g as: [1, ] to[0,] by

If x1 = 1, then x-1 = 0.

Also, g does map from [1, to [0]

Also, g ( g-1(x)) =g(x+1) =x

=

PART B

Schr.oderBernstein Theorem

f: S. T. is injective

g. T S stands for injective

=

[0.] and [0.1]

f: [0?] and [0.1]

f(0) = 2, F(x), =x [ 0,1]

f is injective

g(x), =

Injective clearly g

Schr”oderBernstein’s theorem.

QUESTION 2

Define a map Q: QxQ Q

Step 1: Homomorphum

Justification: [a.b] + (c.d] = (a+c.b+d).

So is hormorphism.

Step 2 is one

Let (a.b. = (c.d).

a-c=0 (b-d=0)

a=c and

Thus, (a.b.= (c.d).

This is how one-one works.

Step 3

This is a justification

Let a+b Q (

a.b QxQ

Also, (a.b. = a+b).

So a.b preimage

From step q3,3, is group homomorphism on-one onto

Thus an isomorphism

Thus, QxQ.+ (QxQ.+) is isomorphic.

QUESTION #3

Va = 1,2,4,57,7.8,8 Is a group

(Z9) = 0,1,2,3,4,5.6,7.8,8 are not part of group but we write carley tables.

It is not necessary to provide formal proof because every entry in field has multiplicative reverse, but Z9 3,6 does not.

Is not field

QUESTION 4

Part A

There are six different digits in the codes.

You have six options to choose from a 6 digit pin.

9 x 9x8x7x6x5= 136080

You have digits ranging from 0 to 9.

The first digit conversion starts from 0 so we need to choose between the remaining 9 digits

Part B

It is allowed to repeat codes that contain only odd numbers, but not digits.

You have many options to choose from a six-digit pin

5x5x5x5x5x5x5x5x5x5x5x5x5x5x5x5x5,5x5x5x5x5x5x5x5x5x5x5x5x5x5x5=15625

We have to choose from 1,3,5,7 and 9, as digits are repeatable and code contains only odd numbers.

Part C

The code can contain at least one odd and at minimum one even number.

The number of options available to select a pin with 6 digits is

5x5x10x10x10x10x10x10x10x10x10x10x10x10x10x10x10x20x10x10x10x10x10x10x10x10x10x10x10x10x10x10x15 = 250000

Part D

These codes contain three odd numbers and three even ones.

You have many options to choose from a six-digit pin

5x4x3x5x4x3=3600

Part E

Case 1: Only one odd number and five even numbers

Case 2: three odd and three equally large numbers

For the first place: 5

2nd place = 5

Third place = 4

For fourth place = 4

For fifth place: 3

6th Place = 3

Multiply 2 because odd numbers can be multiplied by 5x4x3x2

The total number possible ways that 2 can be odd together.

The sum = 5P5 5P1+ (5P3 ),2 + 5.5 (5P2)

Exploring Abstract Geometry With Mathematica.

Chicago: Springer Science & Business Media.

Cayley tables to all semigroups.

Hull: Department of mathematics.