LEARN · EXPLAIN · REVISE

Determine whether x²=3 has a witness in the integers and in the reals.

Read the idea, work independently, then explain what changed.

高一必修第一册(A版).pdf · 1.5 · PDF 33 / printed page 26

TOPIC 01

Universal and existential quantifiers

Build understanding of universal and existential quantifiers through definitions, contrasting cases and justified applications.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

x^2=3

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Keep the two domains separate.

Hint 2

Compare the adjacent integer squares surrounding three.

Worked solution

No integer witness; real witnesses are ±√3.

Checks and common pitfalls: The same predicate can change truth value when its domain changes.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

What must be true before using the main rule for universal and existential quantifiers?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Apply universal and existential quantifiers with explicit conditions.
Scope: Universal means every domain member; existential means at least one witness.
Negation: Swap the quantifier and negate the predicate while keeping the domain.
Close with: conditions → representation → reasoning → check.

Expected reasoning: Universal means every domain member; existential means at least one witness.
Expected reasoning: Swap the quantifier and negate the predicate while keeping the domain.
Expected correction: Examples alone do not prove a universal statement.