• School Mathematics
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• v.18 no.4
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• pp.839-856
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• 2016

In 2009 revision and 2015 revision mathematics national curriculum, 'proof' was moved to high school from middle school in consideration of the cognitive development level of students, and 'proof by contradiction' was stated in the "success criteria of learning contents" of the first year high school subject while it had been not officially introduced in $7^{th}$ and 2007 revision national curriculum. Proof by contradiction is known that it induces a cognitive conflict due to the unique nature of rather assuming the opposite of the statement for proving it. In this article, based on the logical, mathematical and historical analysis of Proof by contradiction, we looked about the introductions and the applications of the current textbooks which had been revised recently, and searched for improvement measures from the viewpoint of discovery, explanation, and consilience. We suggested introducing Proof by contradiction after describing the discovery process earlier, separately but organically describing parts necessary to assume the opposite and parts not necessary, disclosing the relationships with proof by contrapositive, and using the viewpoint of consilience.

• Bulletin of the Korean Mathematical Society
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• v.34 no.3
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• pp.421-436
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• 1997

In this paper we first give a simple proof of the analytic characterization theorems of the operator symbols by using the characterization theorem for white noise functionals. We next give a criterion for the convergence of operators on white noise functionals in terms of their symbols and then use this result to give a proof for the Fock expansion theorem of operators on white noise functionals.

• Journal of the Chungcheong Mathematical Society
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• v.15 no.2
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• pp.67-72
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• 2003

The purpose of this paper is to prove flow induced by a covering map. Lee and Park had studied semiflows induced by a covering map in 1997 [1]. This proof differs from the proof of Lee and Park. Notice that for the proof of this paper, we use the fact that $\mathbb{R}$ is connected space.

• Journal of the Korean Mathematical Society
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• v.47 no.1
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• pp.123-134
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• 2010

The aim of this paper is to provide a proof for a version of the Morse inequalities for manifolds with boundary. Our main results are certainly known to the experts on Morse theory, nevertheless it seems necessary to write down a complete proof for it. Our proof is analytic and is based on the J. Roe account of Witten's approach to Morse Theory.

• Honam Mathematical Journal
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• v.43 no.3
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• pp.403-416
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• 2021

It is known that the rank 2 stably free syzygy module P⁽²⁾ is not free. This algebraic fact was proved analytically, but this remarkable fact still lacks of a simple algebraic proof. The main purpose of this paper is to give a partially algebraic proof by making use of a theorem whose proof is quite topological, and the further properties of the module will be discussed.

• Bulletin of the Korean Mathematical Society
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• v.42 no.4
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• pp.851-853
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• 2005

It is well known that the general term of a sequence can be represented by a linear combination of binomial coefficients. In this paper we give a simple proof for this fact.

• Journal of the Korean Mathematical Society
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• v.47 no.2
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• pp.331-350
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• 2010

We consider the variational problem consisting of minimizing a polyconvex integrand for maps between manifolds. We offer a simple and direct proof of the existence of a minimizing map. The proof is based on Young measures.

• Communications of the Korean Mathematical Society
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• v.25 no.1
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• pp.27-36
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• 2010

We give a simple proof of certain mapping properties of the p-adic Riesz potential and Bessel potential, and the p-adic version of the Sobolev embedding theorem obtained in [6].

• Bulletin of the Korean Mathematical Society
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• v.45 no.2
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• pp.273-275
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• 2008

We will give a short and elementary proof of the existence of infinitely many primes p such that a given positive integer a congruent 3 modulo 4 is a quadratic non-residue modulo p.

• East Asian mathematical journal
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• v.24 no.5
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• pp.465-476
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• 2008

Even though Godel's theorem is remarkable to mathematics teachers, it is not simple to understand the proof in detail. It would be useful for us to understand the basic ideas and the proving process of the proof. In this note, we suggest a proposal for the purpose.