الصفوف 3–5Foundationsمُخطَّط له
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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ستتعلّم
- 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.
تعريف
- 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.
مبرهنة
- 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.
مفاهيم خاطئة شائعة
- 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'.
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اكتشافات اختيارية
- 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.