嬉しそうに、スイカを抱えてこちらに走ってくる長男の姿を･･･ 0034１３２人目の素数さん2019/08/14(水) 14:41:50.08ID:IXi7B7ja To prove an implication “If X, then Y ”, the usual way to do this is to first assume that X is true, and use this (together with whatever other facts and hypotheses you have) to deduce Y . This is still a valid procedure even if X later turns out to be false; the implication does not guarantee anything about the truth of X, and only guarantees the truth of Y conditionally on X first being true. For instance, the following is a valid proof of a true proposition, even though both hypothesis and conclusion of the proposition are false: Proposition A.2.2. If 2 + 2 = 5, then 4 = 10 - 4. Proof. Assume 2+2 = 5. Multiplying both sides by 2, we obtain 4+4 = 10. Subtracting 4 from both sides, we obtain 4 = 10 - 4 as desired. 0035１３２人目の素数さん2019/08/14(水) 14:43:53.88ID:IXi7B7ja>>34
Taoさんは、こんなに懇切丁寧に説明しています。
Taoさんもこのあたりを理解するのに苦しんだと推測されます。
Taoさんは本当に天才なのでしょうか？ 0036１３２人目の素数さん2019/08/14(水) 14:55:15.97ID:IXi7B7ja Here is a short proof which uses implications which are possibly vacuous. A.2. Implication 315 Theorem A.2.4. Suppose that n is an integer. Then n(n + 1) is an even integer. Proof. Since n is an integer, n is even or odd. If n is even, then n(n+1) is also even, since any multiple of an even number is even. If n is odd, then n + 1 is even, which again implies that n(n + 1) is even. Thus in either case n(n + 1) is even, and we are done. Note that this proof relied on two implications: “if n is even, then n(n + 1) is even”, and “if n is odd, then n(n + 1) is even”. Since n cannot be both odd and even, at least one of these implications has a false hypothesis and is therefore vacuous. Nevertheless, both these implications are true, and one needs both of them in order to prove the theorem, because we don’t know in advance whether n is even or odd. And even if we did, it might not be worth the trouble to check it. For instance, as a special case of this theorem we immediately know Corollary A.2.5. Let n = (253+142)?123?(423+198) 342 +538?213. Then n(n + 1) is an even integer. 0037１３２人目の素数さん2019/08/14(水) 14:55:38.28ID:IXi7B7ja In this particular case, one can work out exactly which parity n is - even or odd - and then use only one of the two implications in the above Theorem, discarding the vacuous one. This may seem like it is more efficient, but it is a false economy, because one then has to determine what parity n is, and this requires a bit of effort - more effort than it would take if we had just left both implications, including the vacuous one, in the argument. So, somewhat paradoxically, the inclusion of vacu- ous, false, or otherwise “useless” statements in an argument can actually save you effort in the long run! (I’m not suggesting, of course, that you ought to pack your proofs with lots of time-wasting and irrelevant state- ments; all I’m saying here is that you need not be unduly concerned that some hypotheses in your argument might not be correct, as long as your argument is still structured to give the correct conclusion regardless of whether those hypotheses were true or false.) 0038１３２人目の素数さん2019/08/14(水) 14:57:03.02ID:2WojI+0w タオは数オリ金メダルだよ 天才に決まってるじゃん フィールズ賞よりも数オリのが凄いんだよ 0039１３２人目の素数さん2019/08/14(水) 15:03:00.93ID:vaqFe4Uo ブルバキが最高 0040１３２人目の素数さん2019/08/14(水) 15:04:58.27ID:2WojI+0w ワイは東大の中でも理3なんだよ トップオブ東大なんだよ 高校時代には大学数学終わらせてたし おまえらはどうなの？ 0041１３２人目の素数さん2019/08/14(水) 15:23:34.94ID:lZLNbFS3 NGID:IGatvx/f NGID:2WojI+0w NGID:IXi7B7ja 0042１３２人目の素数さん2019/08/14(水) 16:10:45.32ID:/Fhar0Pm>>41 もう既にNGしてるし、毎回毎回こんな手間が煩わしいからワッチョイ表示をしたいのに現状出来ない
Definition 3.1.4 (Equality of sets). Two sets A and B are equal, A = B, iff every element of A is an element of B and vice versa. To put it another way, A = B if and only if every element x of A belongs also to B, and every element y of B belongs also to A. 0068１３２人目の素数さん2019/08/16(金) 20:46:51.29ID:PHPlyCGu 頭の良すぎる人は、普通の頭の人は何が理解できて何が理解できないかわからないので、 ときどきトンチンカンな説明をします。 つまり、あなたが悪いんです。反省しましょう。 0069１３２人目の素数さん2019/08/16(金) 21:16:59.50 こいつ今その瞬間自分が目の前で意識してる話題についてのレスに対してだけはレス返してくるんだな まんまアスペだわ 0070１３２人目の素数さん2019/08/16(金) 21:18:40.61 ｢僕ちゃん自分のことしか全く見えてません｣感全開やで 0071１３２人目の素数さん2019/08/16(金) 21:22:32.57ID:QIaxdT5W ルベーグ積分のあまり話題に上がらない教科書 マイナーだけあってつまらないものもあるし多くは品切れだがいくつか手に取ると 逆に内容を適宜選択して分厚いテキストよりわかりやすいものもある
