三至五年级Foundations计划中
True, false, and, or, not
Statements that are either true or false, combining them with and, or and not, and testing general claims with examples and counterexamples.
math.found.logic-statements
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前置知识
没有前置要求——从这里开始吧！
你将学会
- Decide whether a simple mathematical statement is true or false and justify the answer.
- Combine statements with and, or and not, reading 'or' inclusively.
- Use a single counterexample to refute an 'every…' claim, and explain why examples alone cannot prove one.
定义
- A statement is a sentence that is either true or false.
- 'P and Q' is true when both are true; 'P or Q' is true when at least one is true; 'not P' is true exactly when P is false.
- A counterexample to 'every x has property P' is a particular x that does not have property P.
定理
- One counterexample is enough to show that a general claim is false.
- Checking many examples can support a general claim but does not prove it.
常见误解
- Believing that several confirming examples prove a statement about all numbers.
- Reading 'or' as exclusive.
- Thinking the opposite of 'all are' is 'none are'; it is 'at least one is not'.
通向
可选发现
- Gödel's incompleteness theorems Is every true statement about whole numbers provable? Gödel showed that any consistent, effectively axiomatised theory containing enough arithmetic leaves some true arithmetic statements unprovable.