Kenneth ChengMathematical modeling
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Decide whether each domain supplies a witness.

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

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

Revisit first: Sufficient and necessary conditions

TOPIC 01

Universal and existential quantifiers

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

What you will be able to explain

  • Apply universal and existential quantifiers with explicit conditions.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Try it. Leave your reasoning visible.

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

01 / Transfer#Worked example

Decide whether each domain supplies a witness.

x2=2;x∈Z or x∈Rx^2=2;\quad x\in\mathbb Z\text{ or }x\in\mathbb R
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Keep the domain explicit and distinguish all values from one witness.
Hint 2
Compare consecutive integer squares around two.
Worked solution

  1. Keep the domain explicit and distinguish all values from one witness.

  2. Calculate or simplify this relation.

    12<2<22;(±2)2=21^2<2<2^2;\quad (\pm\sqrt2)^2=2

  3. An existential statement depends on its domain.

No integer witness; real witnesses are ±√2.

Checks and common pitfalls: An existential statement depends on its domain.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

Focus on one question

Teacher preparation and assessment

Question sequence

  • 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?

Board plan

  • 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.

Anticipated thinking

  • 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.

Assessment checklist

  • 1: identify the givens and required quantity.
  • 1: choose a valid definition, representation or method.
  • 1: present connected, correct reasoning.
  • 1: check conditions and explain the result.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗