1969-03-13 — Page 27

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*2+3 = 1

"x1-1

-1 x (~1) = 1

je 23-1 x 1 and 12-1 x (-1)

34

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1969

現代數學科 (十九)

MODERN MATHEMATICS (19)

Solutions of the problems assigned last week

(1) Prove that the set 1,3,5,9,11,13} mod

oup under multiplication.

Table of multiplication

·5

9

11

11

5

9

5

13

3

21 11 5

13

13 x 13 11 x 11

13 x 13

1

11 x 9 -

11

Test of the associative property:

(3 x 5) x 111 x11

3 x (5 x 11) - 3 x 13 = 11

(3 x 5) x 113 x (5 x 11)

Test of the identity element property:

Obviously, the identity element is 1.

1 x 9 - 9 x 1-

Test of the inverse element property:

That 19, the inverse of is 13, and the inverse of 11 is 9.

Therefore the set is a group under x.

(2) Does the set 1,2,3,4 under multiplication mod

form a group?

Yes, the set forms a group under multiplication mod 5. This can be verified by its multiplication table.

1

3

is the identity element. In each row there

is an element 1. This snows for each element there

is an inverse. The other properties are obvious.

(3) Does the set {1,2,3,4,5} form a group under

multiplication mod 6?

wo, the set under multiplication mod 6 is not a group. This is verified by its multiplication table.

18

2.

The table shows the identity element 1 does not appear in each row. This means some elements in the set have no inverses. These elements are 2, 3 and hence it is not a group."

(4) Do rotations counterclockwise about an origir

through angles of 900, 180°, 270° and 360° form a group under the operation of followed by

Yes, the set of rotation form a group. This is verified by the following table.

Isomorphian

Notice: Ro is 360°

Groups (continued)

18

R270°

Definition: Given G a group with elements at and operation✪, G! a group with elements

etc. and operation 0; then a one-to-one correspor between G and G' defired by anal bb' etc. is an isomorphism if and only if, for each a and b,

a ✪ bea1ob'

Two groups, satisfying the conditions above, are called isomorphic. In other words, two groups are isomorphic if there is one-to-one correspondence between the elements of them and their operations are "preserved" under the correspondence. Hence we can say two groups are isomorphia if they are similar in structure.

Examples

(1) The modular arithmetic mod 4 under addition 1s isomorphic to the group whose elements are

Hence their operations are "preserved".

Notice that the two groups described, have

different operations and different elements, yes they are similar in structure. Therefore, they are isomorphic,

(2) Prove that the group whose elements are 1, >,

11 under the operation of multiplication mod 12

is isomorphic to the group whose elements are

under the operation or substituting

the second factor for z in the first factor. Their multiplication and "substitution tables are given below.

The symol represents the operation or substitution. The one-to-one corres.ondence le

ven by

which indicates the two operations

are "preserved",

8. Some aräunmetic and algebraic goblens

Since any modular arithmetic under Buu Ll Lun 19 an additive group, we can make use of such group to solve some arithmetic questions.

Examples:

(1) It is now y o'clock in the morning, in seven

hours I am going to attend a meeting. At what time am I going to the meeting?

Solution. A clock, as a matter of fast, 19 a modular arithmetic mod 12 under addition. (9 +7) = 4 (mod 12)

dence at 4 o'clock in the afternoon must the time,

(2) 13 days from now eva smali sa to the dentist.

Un what day of the week that shall she go if day is Wednesday?

Solution: The Week system is based on the modular arithmetic mod 7 under addition.

1332 (mod 7)

1.e. The day is Tuesday.

3) It is now March. In 11 months Mr. Wong is going

to the US. In what month is he going? Solution: 11 3 - 2 mod 12)

Answer: He is going to USA. next year in February.

Modular arithmetic may be considered as finite number systems. Algebraic equations over such systems may be solved as follows.

(1) Solve x = 4 (rod 5). Solution:

x 2 = 4; 3 x 3. Hence x 2 or 3 (2) Solve x2 = 2 (moa 7, Solution: 3 x 3 - 2 and 4.

narice

3) Solve x 2 mod 31

Solution: (2x2) x 2 - 1x 2 2, hence x (4) Solve x + 3 1 (pod 5). Solution:

Check: 33

(5) Solva 2x

Solution:

1 (mod 5K

14 (mod 5):

2x

1

2x + 0.4

(-1)

2x = Lix

X *3

-1, i under multiplication. Prove the statement.

Their addition and" multiplication tables are given as below,

2x = 4+

Check : 2x

Ο

I 2 3

X

1

Ο

·2

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1969 OTZ

生物科 (十九 )

(iii) The erector muscle

Ang

廖百参

It is attached to the hair follicle.

When an animal is in a fright or in cold

case; it enables the hair to stand on end. iv) The sebaceous gland

It usually opens into the follicle. It secretes an oily substance called sebum, which makes the skin and the hair soft and water proof.

The mir apilla

At the bottom of the hair follicle is the papilla. It is the region where the new hair is grown. “ (vi) Blood vessels

They are distributed all over the dermis. They supply the skin and hairs with the nutriments and remove waste products.

from the skin. The blood vessels are also to play an important part in the regulation of the body temperature (vii) The nerve fibres and the sensory nem

endings:

Nerves and their endings are distributed. over the dermis. They enable the skin the ability to feel temperature, pressure poin and other external stimuli,

(viii) The fat layer

A region for the storage or rat. It fumtions to prevent heat loss and to protect the tissues from mechanical injury

State the general functions of the skin of a mammal. What functions of the skin do you consider are similar to the functions performed by the epidermis of a leaf of the flowering plant?

The general functions of a mammalian skin are: (a) Protection

The cuticle of the skin, being composed of hard, resistant cells, forms the protective layer of the body to prevent the underlying tissues from drying, from injury am from the invasion of bacteria and acro-organisms. (b) Sensory

in the dermis are found the sensory organs) and nerves for heat, cold, touch and pain sensations

Excretion

The skin is frequently said to have a minor excretory function. The metabolic waste products of the body, such as water, carbon dioxide and dilute urea are discharged by skin. in the form of sweat,

(d) Storage

The fat stored in the deeper part or the dermis. Ce) Vitamin D manufacting

It produces Vitamin D when it is exposed

to the ultra-vislet ray.

(f) Regulation of body temperature

An important function of the skin is the role which it plays in the regulation of the body temperature. This is accomplished by the reactions of its blood vessels and by the evaporation of sweat.whats

Some of the above functions of the skin are

Considered as similar to the functions performed by the epidermis of a leaf. These functions are: again described briefly as the follows:-

(1) To act as a protective layer of the underlying

tissues.

(ii) To control the excess Loss or water.

lii) To provide a thick, external covering for the

internal parts/

(iv) The presence and the function of its oily

secretion. the sebum,

Questions for this week

What is a gland? Illustrate your answer by

reference to the following glands and the functions of their secretiona

(a) Salivary gland

Sweat gland.

(c) Liver":

(d) Pituitary gland

(e) Thyroid gland

(f) Thymus gland

(g) Adrenal gland

(h) Islets of Langerhana

(1) Parathyroid gland

(j) Pineal body

Make a comparison between an enzyme and a hormone.

The one-to-one orrespondence between their elements is given as

1

Exercises Iur une week.

rove that the modular arithmetic mod 3 under addition is isomorphic to the alemer

are the permutation roup whose

• - (123), P. (3 12). - (222) (주)

# 23

under the operation called "followed by denoted by 0. Notice that pog means the permutation p is followed by the permutation c Obviously poq - e

(2) Prove that the set (<,4,6,8} under multiplication

under

mod 10 is isomorphic to the set flot

multiplication mod 10.

13) Bolve

25 (mod 7).

(4) Solve

(mod 1112

what will be the day the week 17 days from now, assuming now is Thursday?

(6) What month of the year will be 37 months from

now, assuming now is March?