TMUA Logic and Proof Techniques | TMUA 逻辑与证明技巧

📚 TMUA Logic and Proof Techniques | TMUA 逻辑与证明技巧

The TMUA (Test of Mathematics for University Admission) Paper 1 contains 20 multiple-choice questions testing mathematical reasoning and logic. The logic component focuses on propositional logic, truth tables, implications, quantifiers, and the ability to evaluate arguments systematically. This article breaks down every logic and proof technique you need, with paired explanations in English and Chinese.

TMUA（大学录取数学测试）第一试卷包含20道选择题，考查数学推理与逻辑。逻辑部分重点考察命题逻辑、真值表、条件语句、量词以及系统评估论证的能力。本文以中英对照的方式，全面解析你需要的每一个逻辑与证明技巧。

1. Understanding the TMUA Logic Syllabus | 理解TMUA逻辑考纲

The logic syllabus for TMUA is deceptively narrow. You are not required to memorise advanced formal logic, but you must be fluent in the core toolkit: recognising propositions, constructing truth tables for compound statements, understanding logical equivalence, translating between English and symbolic notation, and evaluating whether a conclusion follows from given premises. Mastery of these skills is essential for scoring in the upper bands of Paper 1.

TMUA的逻辑考纲看似范围窄小。你并不需要背诵高级形式逻辑，但必须熟练运用核心工具：识别命题、为复合语句构造真值表、理解逻辑等价、在英语与符号表示之间互译，以及判断结论是否从给定前提中推出。掌握这些技能对于在第一试卷中取得高分至关重要。

A key distinction to understand early is between a statement and a logical argument. A statement is a declarative sentence that is either true or false. An argument consists of premises and a conclusion. In TMUA, you will frequently be asked whether an argument is valid — meaning the conclusion must follow necessarily from the premises, regardless of whether the premises are actually true.

早期需要理解的一个重要区别是语句与逻辑论证之间的区别。语句是一个要么为真要么为假的陈述句。论证由前提和结论组成。在TMUA中，你经常会被问到论证是否有效——即结论必须必然地从前提推出，无论前提实际上是否为真。

2. Propositional Logic Essentials | 命题逻辑要点

A proposition is a statement that has a definite truth value — either true (T) or false (F). For example, ‘5 is prime’ is a proposition with truth value T, while ‘x > 3’ is not a proposition unless x has a fixed value. In TMUA, you must identify which sentences are propositions and manipulate them using logical connectives.

命题是具有确定真值（真T或假F）的陈述。例如，“5是质数”是一个真值为T的命题，而“x > 3”在x没有确定值时不是命题。在TMUA中，你需要识别哪些句子是命题，并使用逻辑连接词对其进行操作。

The three fundamental connectives you must know are AND (conjunction), OR (disjunction), and NOT (negation). The conjunction p ∧ q is true only when both p and q are true. The disjunction p ∨ q is true when at least one of p or q is true. The negation ¬p flips the truth value of p. These may seem simple, but TMUA often hides complexity in compound statements that combine all three.

你必须掌握的三个基本连接词是“且”（合取）、“或”（析取）和“非”（否定）。合取 p ∧ q 仅在p和q都为真时为真。析取 p ∨ q 在p或q至少一个为真时为真。否定 ¬p 翻转p的真值。这些看似简单，但TMUA常常将复杂性隐藏在组合全部三个连接词的复合语句中。

3. Truth Tables and Connectives | 真值表与连接词

Truth tables are the most reliable tool for evaluating compound statements. The following table gives the truth values for the standard connectives:

真值表是评估复合语句最可靠的工具。下表给出了标准连接词的真值：

For TMUA, the implication p → q deserves special attention. The only case where p → q is false is when p is true and q is false. This often surprises students: when p is false, the implication is vacuously true regardless of q’s value. You must internalise this truth table completely, as TMUA frequently tests it in multiple-choice questions.

在TMUA中，蕴含 p → q 需要特别关注。p → q 为假的唯一情况是p为真而q为假。这常常让学生感到意外：当p为假时，无论q的值如何，蕴含都是空洞为真。你必须完全内化这个真值表，因为TMUA经常在多选题中考查它。

When building a truth table for a compound statement, work systematically. First list all possible truth assignments for the atomic propositions (2ⁿ rows for n propositions), then compute intermediate sub-expressions column by column. This methodical approach eliminates errors and is acceptable within the 75-minute time limit if practised well.

为复合语句构建真值表时，要系统地操作。首先列出所有原子命题的真值指派（n个命题有2ⁿ行），然后逐列计算中间子表达式。这种有条理的方法能消除错误，如果练习充分，在75分钟的时间限制内是可行的。

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