Kenneth ChengMathematical modeling
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Concept of a set

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

高一必修 第一册(A版).pdf · 1.1 · PDF 9 / printed page 2

TOPIC 01

Concept of a set

Build understanding of concept of a set through definitions, contrasting cases and justified applications.

What you will be able to explain

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

Membership

A set has definite membership and distinct elements.

x∈Ax\in A

Representations

A roster lists members; set-builder notation specifies their domain and condition.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in concept of a set changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

U={1,…,7}; A={1,2,3}; B={3,4,5}. Result={3}. 3 belongs to the result.A: 1,23B: 4,56,7

U={1,…,7}; A={1,2,3}; B={3,4,5}. Result={3}. 3 belongs to the result.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

Count the distinct elements.

A={2,3,2,4}A=\{2,3,2,4\}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Check the membership rule and count distinct objects.
Hint 2
Remove the repeated entry.
Worked solution

  1. Check the membership rule and count distinct objects.

  2. Calculate or simplify this relation.

    A={2,3,4};∣A∣=3A=\{2,3,4\};\quad |A|=3

  3. Order and repetition do not change a set.

The requested value is 3.

Checks and common pitfalls: Order and repetition do not change a set.

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

02 / Standard#Worked example

Count the integers satisfying the condition.

−2<x≤4,x∈Z-2<x\le4,\quad x\in\mathbb Z
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Check the membership rule and count distinct objects.
Hint 2
The first integer is -1.
Worked solution

  1. Check the membership rule and count distinct objects.

  2. Calculate or simplify this relation.

    4−(−1)+1=64-(-1)+1=6

  3. Exclude the open endpoint but include the closed endpoint.

The requested value is 6.

Checks and common pitfalls: Exclude the open endpoint but include the closed endpoint.

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

03 / Transfer#Worked example

Does “interesting books” specify a well-defined set? Repair the description.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Check the membership rule and count distinct objects.
Hint 2
Could two people apply the same rule and disagree?
Worked solution

  1. Check the membership rule and count distinct objects.

  2. Different readers may disagree about membership. Fix a catalogue and a publication year to make membership decidable.

  3. Set membership must be definite rather than a personal preference.

No; use an objective criterion such as books published in a fixed year.

Checks and common pitfalls: Set membership must be definite rather than a personal preference.

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.

04 / Foundation#Your turn

Count the distinct elements.

A={3,4,3,5}A=\{3,4,3,5\}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Check the membership rule and count distinct objects.
Hint 2
Remove the repeated entry.
Worked solution

  1. Check the membership rule and count distinct objects.

  2. Calculate or simplify this relation.

    A={3,4,5};∣A∣=3A=\{3,4,5\};\quad |A|=3

  3. Order and repetition do not change a set.

The requested value is 3.

Checks and common pitfalls: Order and repetition do not change a set.

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

05 / Foundation#Your turn

Count the integers satisfying the condition.

−3<x≤5,x∈Z-3<x\le5,\quad x\in\mathbb Z
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Check the membership rule and count distinct objects.
Hint 2
The first integer is -2.
Worked solution

  1. Check the membership rule and count distinct objects.

  2. Calculate or simplify this relation.

    5−(−2)+1=85-(-2)+1=8

  3. Exclude the open endpoint but include the closed endpoint.

The requested value is 8.

Checks and common pitfalls: Exclude the open endpoint but include the closed endpoint.

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

06 / Foundation#Your turn

Write the real solution set.

(x−3)2=0(x-3)^2=0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Check the membership rule and count distinct objects.
Hint 2
A zero square has zero base.
Worked solution

  1. Check the membership rule and count distinct objects.

  2. Calculate or simplify this relation.

    x=3;A={3}x=3;\quad A=\{3\}

  3. Multiplicity of a root is not the cardinality of its set.

The set is {3}.

Checks and common pitfalls: Multiplicity of a root is not the cardinality of its set.

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.

07 / Foundation#Your turn

Are the two sets equal? Explain.

A={3,4}, B={4,3,3}A=\{3,4\},\ B=\{4,3,3\}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Check the membership rule and count distinct objects.
Hint 2
Compare membership, not position.
Worked solution

  1. Check the membership rule and count distinct objects.

  2. Calculate or simplify this relation.

    A⊆B,B⊆AA\subseteq B,\quad B\subseteq A

  3. Every element of either set occurs in the other.

Yes; the members coincide.

Checks and common pitfalls: Every element of either set occurs in the other.

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.

08 / Standard#Your turn

How many elements are in this set?

A={∅,{0}}A=\{\varnothing,\{0\}\}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Check the membership rule and count distinct objects.
Hint 2
The two objects inside the outer braces differ.
Worked solution

  1. Check the membership rule and count distinct objects.

  2. Calculate or simplify this relation.

    ∣∅∣=0,∣{0}∣=1,∣A∣=2|\varnothing|=0,\quad |\{0\}|=1,\quad |A|=2

  3. An empty set can be an element of a different set.

The requested value is 2.

Checks and common pitfalls: An empty set can be an element of a different set.

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

09 / Standard#Your turn

Give a set-builder description.

A={3,4,5,6}A=\{3,4,5,6\}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Check the membership rule and count distinct objects.
Hint 2
State both the domain and the bounds.
Worked solution

  1. Check the membership rule and count distinct objects.

  2. Calculate or simplify this relation.

    A={x∈Z:3≤x≤6}A=\{x\in\mathbb Z:3\le x\le6\}

  3. An interval without the integer restriction contains additional elements.

Integers from 3 to 6, inclusive.

Checks and common pitfalls: An interval without the integer restriction contains additional elements.

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.

10 / Standard#Your turn

How many elements are in this real solution set?

A={x∈R:x2+1=0}A=\{x\in\mathbb R:x^2+1=0\}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Check whether any real value can satisfy the condition.
Hint 2
A real square cannot be negative.
Worked solution

  1. Check whether any real value can satisfy the condition.

  2. Calculate or simplify this relation.

    x2+1≥1>0;A=∅x^2+1\ge1>0;\quad A=\varnothing

  3. The empty set is different from the singleton containing zero.

The requested value is 0.

Checks and common pitfalls: The empty set is different from the singleton containing zero.

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

11 / Standard#Your turn

Count the distinct elements.

A={4,5,4,6}A=\{4,5,4,6\}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Check the membership rule and count distinct objects.
Hint 2
Remove the repeated entry.
Worked solution

  1. Check the membership rule and count distinct objects.

  2. Calculate or simplify this relation.

    A={4,5,6};∣A∣=3A=\{4,5,6\};\quad |A|=3

  3. Order and repetition do not change a set.

The requested value is 3.

Checks and common pitfalls: Order and repetition do not change a set.

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

12 / Transfer#Your turn

Count the integers satisfying the condition.

−4<x≤6,x∈Z-4<x\le6,\quad x\in\mathbb Z
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Check the membership rule and count distinct objects.
Hint 2
The first integer is -3.
Worked solution

  1. Check the membership rule and count distinct objects.

  2. Calculate or simplify this relation.

    6−(−3)+1=106-(-3)+1=10

  3. Exclude the open endpoint but include the closed endpoint.

The requested value is 10.

Checks and common pitfalls: Exclude the open endpoint but include the closed endpoint.

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

13 / Transfer#Your turn

Write the real solution set.

(x−4)2=0(x-4)^2=0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Check the membership rule and count distinct objects.
Hint 2
A zero square has zero base.
Worked solution

  1. Check the membership rule and count distinct objects.

  2. Calculate or simplify this relation.

    x=4;A={4}x=4;\quad A=\{4\}

  3. Multiplicity of a root is not the cardinality of its set.

The set is {4}.

Checks and common pitfalls: Multiplicity of a root is not the cardinality of its set.

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

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for concept of a set?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Apply concept of a set with explicit conditions.
    • Membership: A set has definite membership and distinct elements.
      x∈Ax\in A
    • Representations: A roster lists members; set-builder notation specifies their domain and condition.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: A set has definite membership and distinct elements.
    • Expected reasoning: A roster lists members; set-builder notation specifies their domain and condition.
    • Expected correction: Repeated entries do not add elements.

    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.

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    Curriculum and source notes ↗