명제 논리를 이용한 고교 과정에 소개된 정리들의 증명
Proofs of theorems from high school courses using propositional logic
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Emphasizing the logic of mathematics in education can be challenging without a clear foundational understanding. By proving trivial theorems in the current high school curriculum using propositional logic, students can gain a deeper understanding of the logical structure of mathematics. In this study, we employed propositional logic to prove core theorems introduced in the high school curriculum. Specifically, we first analyzed the dependencies of propositions involving quantifiers and negated definitions according to formal negation rules. Second, we introduced a rigorous proof method by clarifying the proper placement of prefixes and existential quantifiers in equalities and inverse operations for real and complex numbers. Third, we defined the operations of functions and proved the related theorems using propositional logic. The logical proofs of propositions presented in this study are expected to enhance the sustainability of mathematical research and strengthen learning motivation to pursue studies in science and engineering fields.
Abstract
Emphasizing the logic of mathematics in education can be challenging without a clear foundational understanding. By proving trivial theorems in the current high school curriculum using propositional logic, students can gain a deeper understanding of the logical structure of mathematics. In this study, we employed propositional logic to prove core theorems introduced in the high school curriculum. Specifically, we first analyzed the dependencies of propositions involving quantifiers and negated definitions according to formal negation rules. Second, we introduced a rigorous proof method by clarifying the proper placement of prefixes and existential quantifiers in equalities and inverse operations for real and complex numbers. Third, we defined the operations of functions and proved the related theorems using propositional logic. The logical proofs of propositions presented in this study are expected to enhance the sustainability of mathematical research and strengthen learning motivation to pursue studies in science and engineering fields.