【京大2007】証明の勉強になる良問！命題の真偽判定【整数の性質】

⭐️ "Oshie Math" allows you to ask Hayashi unlimited questions for a fixed monthly fee. You can try asking questions for free once! https://oshiemath.com/ ⭐️ "Hayashi Math Class," a specialized mathematics school where you can receive direct instruction from Hayashi. Please come for a trial lesson and interview! https://hayashi-math.com/ ✅ Official LINE for students applying to top universities: https://lin.ee/lI7n1SJ Subscriber benefits & live streams for applicants ℹ️ Shunsuke Hayashi's profile https://hayashishunsuke.com/profile/ ・Graduated from Sakae Higashi Junior High School → Chikuma High School → University of Tokyo, Faculty of Science, Department of Physics ・Scored 90% on the second-stage mathematics exam at the University of Tokyo, passing the exam as a current student ・2014 Japan Physics Olympiad Gold Medal ・Placed first place in the 2014 University of Tokyo Physics Mock Exam ℹ️ Please note ・The explanations are Shunsuke Hayashi's own and are not official university information. ・Amazon Associates links will be used when introducing books, etc. ★Correction: The √19 written at 02:14 should be √16. This time, we'll be looking at questions about propositions involving integers from the 2007 Kyoto University entrance exam. Determine whether two propositions about the natural number n are true or false. (If true, prove it; if false, explain why.) Those with some proficiency in irrational number calculations and proofs will intuitively predict that proposition p is false. The question now is how to prove it. Since we want to prove that the proposition "both are rational numbers" is false, I think it's simple to assume that both are rational numbers and derive a contradiction. The truth of proposition q is also likely easier to predict, even if not as easy as proposition p. It should intuitively seem like an irrational number. However, because proposition q considers whether the subtraction of two numbers is a rational number, the result of proposition p cannot immediately be used to derive the result of proposition q. (This is because the difference between irrational numbers is not necessarily an irrational number.) It is best to prove it by itself, without forcing it to relate to proposition p. Integer problems, especially those that involve proofs, frequently appear in entrance exams for prestigious universities. If you gain plenty of experience and improve your skills, you'll be able to set yourself apart from other test-takers! ---------- Table of Contents 00:00 2007 Kyoto University Liberal Arts Mathematics [5] 00:30 Review of Rational and Irrational Numbers 01:20 Proposition p: First, Predict Whether It Is True or False 02:37 Proposition p: Assume Both Are Rational 04:07 Proposition p: Lemma and Its Proof 06:26 Proposition p: Proof of Falsehood 08:43 Proposition p: Summary of the Solution 10:49 Proposition q: Prediction of True or False and Points to Note 12:52 Proposition q: Approach to the Proof 14:41 Proposition q: Putting it in the Form n = ... 17:46 Proposition q: Summary of the Solution 19:38 Overall Summary 20:06 Conclusion