등식의 성질과 연산의 정의에 관한 연구 = (A) study on the equality property and the definition of operation

다국어 초록 (Multilingual Abstract)

In this dissertation, we deal with the motive and objectives of 'one-to-one correspondence', especially state in details about the inevitable change from the operation defined between two sets involved algebraic systems to another operation. Also, we studied the condition of 'equivalence relation' accompanied by partition of domain to 'congruence relation' through the study of onto functions.
The school mathematics curriculum should include the inevitable and natural process arising from historical procedure just as it is, according to "Freudenthal's suggestion". Because, students' strong motivation for the curiosity and the aspiration of the mathematiczation is not from some questions on the text book which just deal with various external situation, but from thoughts for seeking solutions of how mathematical states have necessarily developed of itself in internal situation and what the cause and effect relationships are.
In a related matter, the reasons that the properties and operation of equality should be introduced to high school curriculum are;
First, that stage of curriculum should include the way of thinking to solve algebraic-equations in a finite set rather than in the infinite set of integers.
Second, due to this matter, we can figure out the onto-function (: the ring of integers) and make it possible to solve the integer calculation within a possibly finite algebraic system . In this case, if the range were in the finite set, it would be more useful and convenient.
Third, due to this matter solved, should be partitioned. And we can figure out the calculation which is being between the partitions of through the way of integer calculation. We should focus that the equality property is inevitably needed for solving in this process. Further, the embodiment of the equality property is the very operation in this partition.
Finally, the equality property is necessary to define the new operation. Based on this whole understanding, We suggest some subjects which should be introduced to secondary school curriculum.
(1) We need the introduction of clock-arithmetic. Because this makes it easy to understand the usefulness of residue class of integers in this curriculum, and also to learn effectively definitions of operation and the equality property.
(2) The curriculum need Euclidean division algorithm. Because this makes it efficient to calculate ideals in polynomial equation included integer, as well as to figure out the great common divisor in case of having big prime numbers as its factor.
(3) As the major methodology is group isomorphism and ring isomorphism, and then we need "algebraic morphism" which can make the algebraic structures through shifting from the set involved algebraic structures to the other set not involved. This can make the secondary school students understand the value of one-to-one correspondence in the point of global view along with the stream line from the origin of mathematics. Thuswhile following the stream-line of development in mathematics, secondary school students are able to identify the value of onto function as well as that of one to one correspondence from a global point of view.
As a conclusion, transferring an algebraic systems of a domain to it's set function image has given rise to the equality property. From the equality property, the concept "operations" can naturally be originated. Thereby the equality property should be emphasized rather than "operations" in secondary school mathematics.

목차 (Table of Contents)

• Ⅰ. 서론 1
• Ⅱ. 대수학에서 일대일대응함수 3
• 1. 군과 일대일 대응함수 4
• 2. 환과 일대일 대응함수 10
• 가. 이중적분에서 나타나는 일차변환과 행렬의 관계 10
• 나. 체인시스템에서 나타나는 일차변환과 행렬의 관계 13
• 다. 일차변환과 행렬의 일대일 대응 20
• 라. 환과 일대일 대응함수 27
• Ⅲ. Onto함수와 등식의 성질 33
• 1. Onto함수 34
• 가. Onto함수와 분할 38
• 나. Onto함수와 동치관계 41
• 다. Onto함수와 일대일 대응함수 43
• 2. 등식의 성질 45
• 가. 에서 등식의 성질이 성립하기 위한 분할의 조건 46
• 나. 군에서 등식의 성질이 성립하기 위한 분할의 조건 61
• 다. 환에서 등식의 성질이 성립하기 위한 분할의 조건 69
• Ⅳ. Onto Homomorphism의 응용 81
• 1. 디오판토스의 방정식에의 적용 81
• Ⅴ. 중등학교 교육과정의 개선점 88
• 1. 시계산법의 도입 88
• 2. 등식의 성질과 연산의 도입 92
• 가. 중등학교의 연산지도의 개선점 96
• 나. 연산의 근원이 되는 Onto함수의 지도강화 97
• 3. Algebraic morphism의 도입 99
• 가. 연산지도의 새로운 방향 99
• 나. 중등학교에 도입되어 있는 해석적 준동형사상 101
• 다. 중등학교에 도입되어야 할 대수적 준동형사상 109
• Ⅵ. 결 론 118
• 참고문헌 121
• ABSTRACT 123

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