联言判断的真假判断在高中数学必修一学习中运用的实例
Examples of the Application of Truth Value Determination of Conjunctive Judgments in Mathematics Compulsory 1 in High School
李宏 湖北省十堰市郧阳中学
Li Hong Yunyang High School, Shiyan City, Hubei Province
正确运用联言判断及推理的逻辑学基本原理（4）
Proper Use of Conjunctive Judgments and Reasoning
联言判断的五大核心逻辑要素（含义、语言表达形式、结构、真假判断、推理方法）是逻辑学基础。
The Five Core Logical Elements of Conjunctive Judgment—its meaning, linguistic expression forms, structure, truth value determination, and reasoning methods—are fundamental to logic.
展开剩余91%也是 2019 版高中数学必修一构建概念、表述定理、推导性质、判定命题的核心逻辑工具。
They also serve as the core logical tools in the 2019 edition of Mathematics Compulsory 1for constructing concepts, expressing theorems, deriving properties, and evaluating propositions.
联言判断的真假判定规则为：“一假即假，全真才真”。
The rule for determining the truth value of a conjunctive judgment is: "false if any component is false, and true only if all components are true."
“p且q” 复合命题真假判定的逻辑依据。
This principle provides the logical basis for determining the truth value of a compound proposition in the form of "p and q."
也是数学概念判定、性质验证、命题真假判断的核心准则，是数学严谨性的重要体现。
It is also the core criterion for verifying mathematical concepts, validating properties, and judging the truth of propositions, representing a crucial manifestation of mathematical rigor.
一、联言判断真假判断运用的实例
Examples of the Application of Truth Value Determination of Conjunctive Judgments
1. 设p：指数函数的定义域为R（真），q：指数函数的值域为(0,+∞)（真），则p且q为真；
Let p: "The domain of the exponential function is R" (True),
Let q: "The range of the exponential function is (0, +∞)" (True),
Then "p and q" is True.
设p：对数函数的定义域为R（假），q：对数函数过定点(1,0)（真），则p且q为假。
Let p: "The domain of the logarithmic function is R" (False),
Let q: "The logarithmic function passes through the fixed point (1, 0)" (True),
Then "p and q" is False.
2.判定函数f(x)=x2+1是否为偶函数对应联言判断：f(x)定义域关于原点对称且f(−x)=f(x)。
Judging whether the function f(x)=x2 +1 is an even function corresponds to the following conjunctive judgment: the domain of definition of f(x) is symmetric about the origin and f(−x)=f(x).
验证联言支：①定义域为R，关于原点对称（真）；f(−x)=(−x)2+1=x2+1=f(x)（真）。
Verification of Conjunctive Branches: ① The domain of definition is the set of real numbers (R), which is symmetric about the origin (True); f(−x)=(−x)2 +1=x2 +1=f(x) (True).
结论：所有联言支为真，联言判断为真，f(x)是偶函数。
Conclusion: All conjuncts are true, therefore the conjunction is true, and f(x) is an even function.
3.判定函数f(x)=x2（x>0）是否为偶函数对应联言判断：f(x)定义域关于原点对称且f(−x)=f(x)。
Determining Whether the Function f(x) = x² (x > 0) is an Even Function
This corresponds to the conjunctive judgment: The domain of f(x) is symmetric about the origin, and f(-x) = f(x).
验证联言支：①定义域为(0,+∞)，不关于原点对称（假）；②无需验证第二个联言支。
Verification of the conjuncts:
①The domain is (0, +∞), which is not symmetric about the origin (False).
②Verification of the second conjunct is unnecessary.
结论：有一个联言支为假，联言判断为假，f(x)不是偶函数。
Conclusion:Since one conjunct is false, the conjunctive judgment is false. Therefore, f(x) is not an even function.
4.判定 “由所有高颜值的人组成的集体” 是否为集合对应联言判断：该集体满足确定性且互异性且无序性。
Judging whether "the collective composed of all people with high appearance value" constitutes a set corresponds to the following conjunctive judgment: this collective satisfies definiteness, distinctness and non-orderliness.
验证联言支：①“高颜值” 无明确标准，不满足确定性（假）；②无需验证其余联言支。
Verification of Conjunctive Branches: ① There is no clear criterion for "high appearance value", so this collective does not satisfy definiteness (False); ② It is unnecessary to verify the remaining conjunctive branches.
结论：有一个联言支为假，联言判断为假，该集体不是集合。
Conclusion: If one of the conjunctive branches is false, the conjunctive judgment is false, and this collective is not a set.
二、联言判断的真假判断：
Truth Value of Conjunctive Judgment
联言判断是断定对象的几种情况都（同时）存在的判断。
A conjunctive judgment asserts that multiple conditions of an object coexist simultaneously.
因此，当且仅当，组成它的各个联言支都是真的，这个联言判断才是真的。也就是说，它要求各个联言支的断定都要与实际相符合。
Therefore, it is true if and only if every conjunct is true. This requires each conjunct to align with reality.
如果有一个联言支是假的，这个联言判断就是假的。
If any conjunct is false, the entire conjunctive judgment is false.
联言支同真，联言支之间有联系，不能割裂或矛盾。
Key: All conjuncts must be true and logically related; isolated or contradictory conjuncts invalidate the judgment.
三、“条件必须全部满足”的逻辑基础
The Logical Foundation of "All Conditions Must Be Met"
真假规则：当且仅当所有联言支为真时，联言判断才为真；一个联言支为假，整个判断即假。
Truth Rule:A conjunctive judgment is true if and only if all its conjuncts are true; if even one conjunct is false, the entire judgment is false.
教材体现：这个规则是数学推理中“条件必须全部满足”的逻辑基础。
Manifestation in the Textbook:This rule serves as the logical foundation for the principle that "all conditions must be met"in mathematical reasoning.
在定理应用中：定理“如果 a > b 且 c > 0，那么 ac > bc。”
Application in Theorems:For the theorem: "If a > b and c > 0, then ac > bc."
要应用此定理得出结论，必须确保前提“a > b” 和 “c > 0” 同时为真。若c = 0或c < 0，则其中一个支为假，整个前提（联言判断）为假，定理不适用。
To apply this theorem and draw a conclusion, it is essential to ensure that both premises, "a > b" and "c > 0," are simultaneously true. If c = 0 or c < 0, then one of the conjuncts is false, making the entire premise (a conjunctive judgment) false, and the theorem does not apply.
在判断命题真假时：判断命题“√2（根号2）是无理数 且是实数”的真假。
When judging the truth or falsity of a proposition: Judge the truth or falsity of the proposition "√2 is an irrational number and a real number".
因为两个联言支（√2是无理数，√2是实数）都真，所以该联言判断为真。反之，若有一个支为假，则整个命题为假。
Since both of the two conjunctive branches (√2 is an irrational number, √2 is a real number) are true, the conjunctive judgment is true. On the contrary, if one of the branches is false, the entire proposition is false.
Explorer of Innovative Thinking
Author: Li Hong
Yunyang High School, Shiyan City, Hubei Province
January 27配资公司最靠谱三个平台, 2026
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