真二第張六第日八十月正年酉己醫夏 WAH KIU YAT PO
報日僑華
四明星日六月三年九六九一公年八十五國民中育教僑華
英文中學會考 【題預習
Example
1969
داران
學會考試題預習
現代數學科
Prove The arithmetic mod 5 under multiplication:
(十八) 李義
when sero is deleted, constitutes a group. Proof: Table of multiplication
生物科
(十八)
MODERN MATHEMATICS (181
BIOLOGY (18)
Vit Groups (Continued)
Theorem about "group
We are to prove some theorems about groups
the use of the definition of group, rule of substitution, and postulates of equality. Theorem 1.
For any element a, x, y in a group, I mus then X-X where "ot denotes the operation. Proof :
ax Boy
108 109
Thearen IL
If a implies adc Proof
800 = 200
b
a0c boo
Assumption
Inverse element property of a group." Associative property of a group. inverse element property of a groud Identity element property of a group.
c are elements in a group, then asb boo
keflexive property of equality
Assumption
Theorem 111
Rule of substitution"
In a group, the left identity 1 1s e iso thi Fight identity. That is ica a implies aci Proof
Let a
But
be the left inverse of a..
Theorem IV:
Inverse element property or a group.
neflexive property of C equality
a oa101 - Associative property or
e: group.
inverse element property or group and identity element. property of a group
Inverse element property oI group.
Theorem i
ransitive property equality.
In a group, a ser inverse element a also a right inverse element
That 19
Proof
ola on l)ulatoa) Ja
seocitive proper
0
2
3
4:
0
0
0
0:
}
0
2:
3.
Li
2
-0
2
1
3
0
3-
1
2:
0
2
Jimilar to example, the operation is closed associative, and commutative. For each lemem, except 0, there is an inverse element; and there exists the identity element 1. Hence the set 10.1.2.3.47 is a group under multiplication.
Example 3.
Frove that the aritmetic mod 4: 2s not a group under multiplication, even 0 is deleted.
lable of: Hnitiplication
Q. 1. 2 3
0.
0 0 Q 0.
1
•O
1
2
3
2. 0 2:
0 2
.0
3
2
1
The operation is closed, and associative. There is the identity element 1, but not every element has an inverse. 2 has no inverse. Therefore the set 0,1,2,3 is not a group under multiplication,
It should be noted that every modular- srithmetic is a group under addition. However 15 18. a group, under multiplication when the mod is prime and 0 is deleted. But it is not a group under multiplication if the mod is even or composite
Example 4
Let AABC be an equilateral elements we have
group
8. a rotation in the plane through 120, about
the centre of the A
= a rotation in the plane through 240“.
the centre of the A
a rotation in the plane through 360, about the centre of the A
AA flip in space about the altitude.
from A
A flip in space about the from B
A flip in space about the from C
answer to the questions of last week:-
1. On the line at the right of each statement, plaçe
the number preceding the word or expression that best completes the statement.
a plant normone that stimulates growon or plante is (1) cortisone (2) insulin (3) auxın (4) toxin (5) pepsin.
1. (3)
2. Vitamin D in the diet prevents (1) haemophilia
bellazra (3) rickets (4) scurvy (5) sterility
・2 (3)
The eggs of a frog are formed in the (1) overy (12) oviduct (3) sperm duct (4) teates (5) uterus.
3. (1)
The condition, known as cretinism results from 8 lack of (1) calcium (2) thyroxin (3) Vitamin (1) Vitamin A (5) adrenaline)
(2) The placenta is an important structure in the reproduction of (1) amphibians (2) birds (3) reptiles (1) mammals (5) insects.
5. (4)
ine egg or a fish is larger than the sperm because the egg contains (1) yolk (2) jelly coát 13) embryo (1) nucleus. (5) oil.
A disease transmitted by mosquitoes 18. (1) rheumtic fever (2) typhoid fever () tronus fever (4) yellow fever (5) diphtheria
7. (4)
VUGArches mi away from a bright light. This behavior is an example of negative (1) chemotropism (2) geotropism (3) hydrotropism (4) phototropism (5) thermonasty.
14
Anema may be prevented by including in the diet adequate amounts of (1) calcium (2) iron (3) Vitamin A (4) Vitamin K (5) Vitamin D.
9. (2)
10. equal weights or the following substances
are oxidized in the body, the largest number of calories 18 produced by. (1) carbohydrates. (2) fats (3) proteins (4) Vitamins (5) sugars.
10. (2)
nydrotropism is a response to (1) gravity (2) heat (3) light () chemical subetance (5) moisture.
11. (5)
12.
A CUIU-D LOOg animi with lungs and a scaly sk is (1) an amphibian (2) a bird (3) a l (4) a reptile (5) a fish.
12. (4)
ola da “
Theorem
Theorem V
Rule of substitution snd associative property Theorem I1L.
If a ana o are elements in a group, then there exists 8* unique element x = a-Lob such that aux Proof
As sune a ox
-1
da cx
Assumption.
Inverse element property and rule of substitution.
Associative property of. group.
inverse e Leirent property
group.
Identity element
More examples of group
of a group
he shall verify here that a II moduler arithmetic under addition constitute groups, but no modula
arithmetic under multiplication constitutes
Then R, R., A, B, Gj forms a group under the operation of followed by". The elements are shown more precisely as the diagrams shown. below.
A..
ing Smallest feathers on a bird's body are the (1) down feathers (2) quill feather a (3) cont.aur feathers (4) coverts (5) 11oplumes
13. (5)
anica item aces not belong to the group (1) housefly (2) termite (3 honey bee (4) Scorpion (5) Grasshopper
∙14 (4)
A 11sn detects vibrations in the water by means of its (1) Fins (27 lateral lines (3) eres (4) swin-bladder (5) cills.
15.
The disease called beri-bera as due to (1) insufficient secretion of bile (2) excess secretion of gastric juice (3) absence of proteins in the diet (4) absence of Vitamin E in the diet⠀⠀ 14) absence of Vitamin B in the diet
16.
group, The latter is due to the fact trat zero and some other numbers have no multiplicative inverse. We shall also verify that the set of all symmetries of any gnometric figure constitutes a group
Example 1
Prove: Arithmetic moq 3 under addition is a group Proof: Fable of addition
O ·1 2
0
1 2
1
1-
0
-2
2
0
The table shows it is closed under addition, for all the elements are belonging to the set. lo. 1, 2
The operation is associative: (0 - 1) - 2 0+ (1 + 2), (2) ► 1) + 1 = 2 + (1 + 1), and so on.
The identity element is 0.
For each element there is an inverse. This can be asserted readily, because in each row of the. table there is a auro, That is 0 is 0. 1 is and 2-19 1
If a line segment is drawn from the upper lart corner to the lower right corner of the table, you see the other elements not in the diagonal are symmetrical with respect to the diagonal. Hence the operation is commutative,
Therefore
addition.
the set f0.1.21 is a group under
R
و به
B
The operation followed by "0" is closed associative and there is the element of identity,
Since R occurs in each row,
there is an inverse
each element. For example H
is Roy 19 R 19 A, B is B and so on. There the set of the elements under "O" is a group.
6. Axercises
(1) Frove set {1,3,5,9,11, 13 med 14 is a group.
under multiplication,
(2) forn
multiplication mod 5
a group under
(3) Does the set {1,2,3,4,5) form a group under
multiplication mod 67
14) Does rotation counterclockwise about an origin through angles of 900, 180°. 270° and 360° form a group?
Reference: "Basic Concepts of New Mathematics! by Y. Lee. Essential Press, 6-C Nelson St. Kowloon.
7. Which roca substance helps to prevent night
blindness and other eye diseases?
(1) proteins (2) Vitamin A ) Vitamin B (4) Vitamin C (5) fats
17. (2)
Choose the item which is the least useful ir helping plants to disperse their seeds or fruits. (1) Maman beings (2) Birds (3) Insects (4) water (5) wind.
13. (3)
19. Mmeral salts in the soil can only be absorbed
by root-hairs if they
(1) contain nitrogen
(2) are dissolved in water
3) are insoluble in water
are mixed with lime
(5) are decomposed into other elements
20. Locomotion in tadpole is mainly brought about by: (1) muscular tail (2) legs external zills (4) sucker (5) fins.
Ansi
2011
(a) With the aid of a fully labelled diagram,
describe clearly, the structure of the bar of 3 Named mammal. (b) Part of the ear is concerned with the function of balance of the body. Describe fully this part of the ear and show how does it work?
The diagram below is a partly section of the buran
ea
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