TMUA 逻辑证明笔记 阅读顺序 条件句 逆否
TMUA 是 UAT-UK 为数学、统计、计算机与经济类学位设置的入学考试，两卷各 20 道选择题、各 75 分钟，全程不能用计算器。内容说明把考点切成两块：Section 1 是两卷共用的数学知识，Section 2 只对 Paper 2 生效，讲的是逻辑与证明。官方另有一份 Notes on Logic and Proof，专门覆盖 Section 2 的条目。
UAT-UK 官网给的准备建议只有三样：读内容说明、读那份逻辑与证明笔记、按计时条件做完练习卷。很多学生以为笔记要逐页啃完才算准备好，于是三份材料同时打开乱翻，笔记读完了却说不清 Paper 2 到底考什么。这份笔记怎么用、按什么顺序用，直接决定 Paper 2 的复习效率。
The UAT-UK website prescribes a short reading order: the Content Specification first, then the Notes on Logic and Proof, then a practice test under timed conditions. Section 2 of the specification is the scope of Paper 2 alone, and it lists only three blocks of items, namely Arg for the logic of arguments, Prf for proof types, and Err for finding faults in purported proofs. The official notes exist to fix the meaning of that vocabulary rather than to supply exercises, and no examiner awards marks for having read them. A candidate who drills questions first tends to explain every wrong answer as carelessness, because there is no item list to trace the error back to.
官方三件套的阅读顺序：内容说明、逻辑与证明笔记、计时真题
UAT-UK 官网把材料排得很清楚：先读 Content Specification 里的 Section 2 清单，确认 Paper 2 只考 Arg、Prf、Err 三组条目；再读 Notes on Logic and Proof，把每条术语的理解落到具体例子上；最后打开练习卷，按 75 分钟做完 20 题。这份笔记本身没有习题也没有答案，只读不练，等于记住了一串词。顺序反了也会出问题，先刷题再看笔记，错题原因往往归结不到具体条目上。笔记的作用是给你一套判据，用来给错题贴标签。读完一条就回内容说明里找到对应的编号，把两处的说法对一遍。
The prescribed order runs from the item list to the notes and finally to timed practice, and each stage answers a different question. The specification tells you what may be assessed, the notes tell you what the terms mean, and the practice test tells you which of them you can actually handle at speed. Since the booklet carries no exercises and no answers, reading it alone leaves you with vocabulary rather than technique. Working backwards from questions to notes tends to leave a wrong answer unexplained, because nothing in the reading has flagged the item it belongs to.
Paper 1 与 Paper 2 的分工：Section 1 与 Section 2
两卷共用同一份数学知识清单，区别在第二卷还要吃下 Section 2。Paper 1 只考 Section 1，题目把 AS 阶段的纯数内容放进陌生情境里，考的是辨认该用哪个工具；Paper 2 在 Section 1 的基础上加考推理，要求判断一串推导是否站得住。两卷各 20 题、各 75 分钟，连续作答，合计 2 小时 30 分钟。计算器与字典都不允许使用，也没有公式表，公式全靠自己记得。答错不倒扣，所以每道题都要填上答案，留空等于白送一分。很多学生把 Paper 2 当成 Paper 1 的续集来准备，题型一换就找不到节奏。
Both papers draw on the same mathematics, but Paper 2 charges a second toll: it tests Sections 1 and 2 together, so reasoning sits on top of content. Paper 1 places familiar pure mathematics inside unfamiliar settings and asks which tool applies; Paper 2 asks whether a chain of reasoning holds at all. Forty multiple-choice questions in total, twenty per paper at seventy-five minutes each, which adds up to two and a half hours back to back. No calculator, no dictionary and no formulae booklet, so recall replaces lookup and every unanswered question is simply a mark discarded.
那份笔记的行文顺序：术语、证明骨架、找出错误
笔记的行文顺序就是它希望你使用的顺序。开篇处理 Arg 组，从 true、false 和 and、or、not 讲起，再把 if 的四种写法逐一拆开；中间一章处理 Prf 组，给四种证明类型配例子；最后一章处理 Err 组，专门展示一份看似成立的伪证错在哪一步。按这个顺序读，每一步都踩在前一步的词汇上；跳着读会卡在 necessary 和 sufficient 这类术语上，因为它们要靠前面的条件句才能定义清楚。常见的错误是只读完第一章就冲去刷 Paper 2，而 Prf 与 Err 两组的题占了大半。读完一章就回内容说明核对条目号，确认自己能用一句话说清每条的含义。
The booklet's own order is the order to use it in, and the three chapters build on one another. It opens with the Arg items, moving from true and false through and, or and not, and then separating the four ways of writing a conditional; the middle chapter walks through the proof types with worked examples; the closing chapter dissects purported proofs line by line to show where a faulty step sits. Working through it in that sequence means each page stands on the vocabulary of the last. Skipping ahead stalls on words such as necessary and sufficient, which only settle once the conditional forms are in place, and finishing the first chapter says nothing about Prf or Err, where most of the marks live.
四种条件句的指向：if A then B、A if B、A only if B、A iff B
if A then B 说的是 A 推出 B；A if B 方向相反，是 B 推出 A；A only if B 又回到 A 推出 B，B 成了 A 成立的必要条件；A if and only if B 两头都成立，两个命题同生共死。官方内容说明写明，考生不需要使用符号写法，也不要求画真值表，题目全部用文字陈述。做题时把每个条件句翻译成箭头，方向就不会反。把 only if 读成 if 是 Paper 2 里丢分最快的一种误读。很多学生以为 A only if B 等于 B 推出 A，把它与前两种形式混在一起。A only if B 的逆否写法是 not B 则 not A，这条等价关系可以用来做检验。
Four conditional forms, two directions. If A then B points from A to B; A if B reverses that and points from B to A; A only if B returns to A implying B, which makes B a condition that A cannot do without; A if and only if B runs both ways at once. The specification is explicit that symbolic notation is not required and formal truth tables are not expected, so every statement arrives in words. Turning each sentence into an arrow before choosing an option is the cheapest habit available on the paper, because reading only if as if sends the reasoning in the opposite direction.
原命题、逆命题与逆否命题的真值关系
一个命题为真，它的逆否命题必然为真，它的逆命题则不一定。if A then B 的逆否是 if not B then not A，两者同真同假；逆命题 if B then A 只是换了方向，真值可以完全不同。A only if B 的逆否是 not B only if not A，等价关系同样成立。检验办法很直接：否定结论，看能不能推出与已知条件矛盾的结果，能推出来就说明原命题更可信。常见的错误是拿逆命题当原命题使用，把充分条件当成必要条件。笔记里那条由 sin A 等于 sin B 不足以推出 A 等于 B 的提醒，说的就是这个方向问题。
A statement and its contrapositive always share a truth value, while a statement and its converse usually do not. Negate and swap, and the result may be relied on; swap alone and you hold a claim that merely points the other way. For A only if B the contrapositive reads not B only if not A, and the equivalence holds there as well. A practical habit is to assume the conclusion fails and see whether the known conditions break under that assumption, since the route often settles a doubtful option in a single line.
necessary 与 sufficient：必要条件与充分条件的方向
necessary 和 sufficient 在 Paper 2 里出现频率很高，很多学生把它们当成同义词。充分条件是够用的条件，满足它，结论必然成立。必要条件是缺不得的条件，结论成立时它必须已经成立。A only if B 里的 B 是必要条件，if A then B 里的 A 是充分条件，箭头永远从充分指向必要。题目常给两个条件让考生判断关系，判断标准就是箭头的方向对不对。常见的错误是把必要条件当成充分条件，看到 B 成立就断言 A 成立。把两个词换成中文的够用与缺不得，判断会快很多。
Necessary and sufficient describe the two ends of the same arrow. A sufficient condition is strong enough on its own, so meeting it forces the conclusion through; a necessary condition cannot be missing, so if the conclusion holds then the condition must already hold. In A only if B the letter B is necessary, while in if A then B the letter A is sufficient. Questions usually hand you two conditions and ask which relation holds, and the direction of the arrow decides the answer every time.
for all、for some 与 there exists 的量词否定
Arg 组还有两条关于量词的条目。for all 是全称说法，一个反例就能把它推翻；for some 在官方口径下等于 for at least one，只要求至少存在一个例子；there exists 与 for some 同义。否定时量词要一起换，对所有 x 都成立的否定是存在某个 x 不成立，反过来也一样。常见的错误是把 for some 理解成日常英语的有些但不是全部，官方特意写明它只表示至少一个。做题时先把量词和结论一起否定，再回头看选项。反例在选择题里非常好用，代入一个数就能筛掉一批选项。
Two more items govern quantity rather than connection. For all is a universal claim, and a single counterexample destroys it; for some means for at least one in the official reading, so one instance makes the claim true; there exists says the same thing in different words. Negating one of these swaps the quantifier along with the claim, so the denial of a for all statement is a there exists statement about failure. Many students import the everyday sense of some as some but not all, and that slip removes an option they could have eliminated by testing a number.
官方 Prf1 的四种证明类型与反例的用途
官方 Prf1 只列出四种证明类型：直接演绎、分情况讨论、反证法、举反例推翻。直接演绎从已知条件一路推到结论；分情况讨论把对象切成互斥的几类分别处理，奇偶讨论是最常见的例子；反证法先假设结论不成立，再推出矛盾；举反例只能用来推翻全称命题，不能用来证明任何东西。笔记与内容说明里都没有数学归纳法，这一点要注意：A-Level 阶段学着顺手，但它不在 Paper 2 的条目里。把归纳法当主力来练，等于把时间花在不考的地方。
The official list of proof types is short: direct deduction, proof by cases, proof by contradiction, and disproof by counterexample. Direct deduction walks from what is given to what is claimed; cases split the objects into disjoint groups and settle each one, with even and odd the standard example; contradiction assumes the claim fails and drives that assumption into conflict with a known fact; a counterexample only ever demolishes a claim and never establishes one. Mathematical induction appears nowhere in the specification or in the notes, so drilling it for this paper is time spent where nothing is examined.
Prf3 与 Prf4：从小例子猜结论、把乱序语句排成证明
Prf3 要求从小例子里猜出规律并给出理由，Prf4 要求把一组乱序语句排成正确顺序，还原出一份证明。两条考的是同一件事的两面：前者从数据走到猜想，后者从猜想走回数据。做 Prf3 的题，先算两三个特例，记下规律，再说明为什么它对一般情形成立，只写规律不写理由拿不到分。做 Prf4 的题，先找结论句，再找唯一以已知条件开头的那句，最后看哪句只能接在前一句后面。很多学生会从头读选项，读到最后才发现拼不起来。先找结论再找起点，比顺着读快得多。
Two items ask for opposite journeys. Prf3 wants a conjecture drawn from small cases together with a justification: compute two or three instances, state the pattern, then say why it must hold in general, since the pattern by itself earns nothing. Prf4 hands you the same statements in the wrong order and asks for the sequence that turns them into a proof. Locate the conclusion first, then the only line that can start from the given data, and the remaining links tend to fall into place without further search.
Err1 与 Err2：识别伪证里的无效推断
Err 组的两条专门盯伪证。Err1 要求在写得很像样的一段推导里找出错误的那一步，Err2 要求认出常见的无效推断，官方笔记举的两个例子是：由 a 的平方等于 b 的平方就断言 a 等于 b，以及由 sin A 等于 sin B 就断言 A 等于 B。两处的毛病一样，都是把单向的结论当成可以反推的结论。做题时逐行检查每步变换是否可逆，尤其盯两边平方、开方、约去公因式这几种操作。阅卷时看到的错误，多半不是算错，而是把不可逆的一步当成了可逆的一步。
The two Err items aim at arguments that look respectable on the page. Err1 asks you to find the single faulty step in a printed proof, while Err2 asks you to recognise invalid deductions by pattern; the official examples are the leap from a squared equals b squared to a equals b, and the leap from sin A equals sin B to A equals B. Both failures are failures of reversibility, treating a one-way implication as though it could be read backwards. Squaring, taking roots and cancelling common factors are the operations worth inspecting line by line.
TMUA 评分口径：中位数 4.5 与第 90 百分位 7.0
UAT-UK 官网说明，考生拿到的是一个总分，标尺从 1.0 到 9.0，保留一位小数，没有及格线。评分的锚点是固定设计出来的：中位数定在 4.5，第 90 百分位定在 7.0，官方在准备页里把这两个数写得很直接。标尺在 2024 年改版过，UAT-UK 明确提醒不要拿旧年份的分数直接比较，因为等值方法换了。剑桥时期的旧标尺用的是另一组锚点。所以听到别人说 6.5 是好分数时，先确认他说的是哪一年的分。分数只是申请材料的一部分，课程页怎么写才决定你要不要考这一场。
UAT-UK reports one overall score on a scale running from 1.0 to 9.0, quoted to one decimal place, with no pass mark attached. The anchors are fixed by design, since the median sits at 4.5 and the 90th percentile at 7.0, and the official preparation page states both plainly. The scale was revised for 2024, and the provider warns against comparing scores across that change because the equating method differs; the earlier Cambridge-run scale used a different set of anchors. When somebody describes a score as good, the year of the sitting matters as much as the number.
十月场与一月场的选择与中国内地港澳考位限制
2026 年 10 月场次在 12 日至 16 日，2027 年 1 月场次在 4 日至 8 日，两场都可用于 2027 年入学申请。剑桥与牛津的申请人通常必须参加十月场，只有少数一月截止的成熟学院或基础年项目例外。UAT-UK 官网另有一条针对中国内地、澳门与香港考生的规定：十月场里 TMUA 只能订 15 日至 16 日这两天。报名时间与截止时间每轮都在变，以 UAT-UK 官网 Key Dates 页为准。进考场后的做题顺序，建议按两分钟一轮推进：先把一眼看出思路的题做完，把卡住的题标上编号，回头再算。
The October 2026 window runs from the 12th to the 16th and the January 2027 window from the 4th to the 8th, and either sitting can serve an application for 2027 entry. Cambridge and Oxford applicants are normally required to sit in October, with mature colleges and foundation-year programmes the narrow exceptions. UAT-UK also restricts candidates sitting in mainland China, Macau and Hong Kong to the 15th and 16th of October for the TMUA. Booking dates shift each cycle, so the Key Dates page on the UAT-UK site is the only reliable reference, and inside the exam a two-minute sweep through the paper beats grinding one question while the clock runs.
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