真二第張六第
日一十月正年西己腦
WAH KIU YAT PO
1969
.ស
現代數學科
(TE)
李義
MODERN MATHEMATICS (17)
113, Solutions
(b) y •
Given
• y}
Doms in the set of real numbers the set of real numbers.
given
↑ = {(x,y)]y » *}
Doma in the set of real numbers
Range - the set of positive real numbers
bx2
=√16 - wx2
= {(x,y) { y = √16
Doma in a
·{x | -2 = x ≤2}
given
報日僑華
of them have properties in common, and it is
more economical for us to study only a few each of which represents a set of mathematical systems that are common in their properties. In this lesson, we are to discuss a particular type of mathematical systems which have the following properties in
common (1) the set is closed under one operation: 12) the operation is associative; (3) there is an identity element; (4) for each element in the set there is an inverse. Such kind of a mathematical Bystem is called a group..
A more precise definition of a group is given below,
group 19 set of elements G; together with one operation" satisfying the following conditior La G is closed under operation "0"
(b) For any elements a, b, c in 6, golpoc) - laobjo
c. There is the element 1 in such that for aris
1 0207
(d) for every element in G, there is an element
in G such that
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下年度小學教育財政預算
(*) x2
Dors in
Range
the set of positive real numbers or zero.
the set of positive real numbers or zero.
1x1245
Domain
{y 10 < y < oo]
givei.
3х
What are mentioned above are considered es postulates for groups also. Upon these postulates
some theorems of groups can be proved. The 4 postulates may be re-stated as the closure postulate, the
associative postulate, the identity element postulate.
and the inverse element postulate. A system may have
other properties besides these, but it is still called a group.
zemerks:
(a) The operation-"0" may represent operation of
whatever kind. In some cases it may be called multiplication, then the group is called multiplicative group. In other cases it may be addition, then the group is called additive group. In still other cases, it may be operation called pre-multiply" or "followed by", The commutativity is not required by the definition of a group. "Lf the commutative property holds for a group, then it is called a commutative group or Abelian group in honour of: the worwegian mathematician, Niel Abel
examples of groups
A mathematical system can be devised for a four-minute clock whose face has only four numbers: 0, 1, 2, and 3. Now let us test whether the four postulates are satisfied with respect to each of the two operations, addition and multiplication..
Let us form the add 12102 taché ang multiplication table of the system as below,
with
respect to the
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vertical axis
The symmetry corresponds to a rotation of 180° about the axis, denoted by . The symmetry with respect to the horizontal axis corresponds to a rotation of 180° about the axis, denoted by H. The symmetry with respect to the centre corresponds to a 180 rotation in the plane of rectangle, denoted by R. The last symmetry means a rotation for completeness, denoted by I. The following diagrams show the meanings of these rotation.
....
{(x,y) La
3x
0 I 2
Domain hange
the set of real numbers the set of real numbers.
0:
0.
1 2
gaven
2 3 O
3-
OWNT
2
•1
MOHN
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ง
2
0
03
Hawolw
Domain
{(x,y) | y=√= }
the set of positive real numbers Kange the set of positive real numbers. The inverse of f is not a function,.
given
Lomin >
{y 10 = y < 2 }
nange
(a) y = √→
q~] =√(x, y) { y = 3√
Domain the sit of 11.
Range the set of all portav
Domain
15.
real mater′′9.
number a.
the set positive real numbers or zero. Hange the set of positive real numbers.
{-2,2}, 1-2(0) - Ø, ƒ ̄(10) - {3,-3},
£ (-5). - Ø
g* (1) -{x / x = =
+217, where n & 2}
Test for issociative property:
Addition
(172) 3 3+3 = 2
1+ (2+3)= 1+1 - 2
(1+2)+3=1+(2+3);
Multiplication
(12)x3 = 23 = 2
1x (2x3) = 1x2 =
(1x2)x3 = x(2x3)
The other combinations of any three of the elements 0,1,2,3 will justify the associative property is well..
Test for closure property:
This is obvious from the tables
Test for the identity elemant propertý :
Addition
0x = x, where
x is any element in
Multiplication
1 x x < x, where x is any element in G.
This shows that the additive identity, element is 0 and multiplicative identity element is 1.
Test for the inverse element property: From the addition table you see in each ro there is the element 0. This verifies that there is en inverse element for each element in G. For example, the inverse of 2.
2 is 2, of 3 is of 1 is 3. Notice that the additive inverse of an element a is denoted by -a, hence -2 is 2,
3 is 1,-1 is 3. In the multiplication table, you see that 1 does not occur in every row Therefore, there is no multiplicative inverse for each element..
We conclude that the set G, where & ► {0,1,2,3 is a group with respect to addition, but it is not group with respect to multiplication.
(b) Me shall consider a simple mathematical system
containing not numbers, but other objects called: rotations under the operation called "followed. by, denoted by "x".
Given any rectangle ABCD, there are 4 symmetries, namely a symmetry with respect to a vertical axis, a symmetry with respect to a horizontal axis, a symmetry with respect to the centre, and a symmetry with respect to e rotation of 360 in the plane of the rect angle. Each symmetry corresponds to a rotation leaving the rectangle with its general appearance unaltered,
A
Below is a "followed by table. The "followed by" "operation means to perform one rotation, then it is followed by another rotation, Thus V * H mens to perform the rotation V, then the rotation H.
The cable shows the operation * is closed, commutative, and associative. There is the identity element I, and for each element there is an inverse. For example, the inverse of 1 is H, of His H, of V is V, of R is R. Hence we conclude the set T. « {I, V, H, R} is a commutative group under the operation "followed by.".
Show that û = 11, is a group.
-1, -1 under multiplication
Table of the operation
(2) = { f
Grups defined
A mathematical system is described by specifying same set of numbers or other objects, and by defining one or more operations for the objects in that set, together with some basic properties of these operations. No doubt, mathematical systems are in great variety, It is impossible for us to study them all. Fortunately
The table shows the set under multiplication satisfy all the 4 condicions required. Besides, the operation is commutative. Hence the set under "x" in
commutative group.