Kenneth ChengMathematical modeling
← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Basic relations between sets

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

高一必修 第一册(A版).pdf · 1.2 · PDF 14 / printed page 7

Revisit first: Concept of a set

TOPIC 01

Basic relations between sets

Build understanding of basic relations between sets through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Apply basic relations between sets with explicit conditions.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Inclusion

Every member of A must belong to B.

A⊆BA\subseteq B

Equality

Equality requires inclusion in both directions.

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 basic relations between sets 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

How many subsets does A have?

∣A∣=3|A|=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Apply the element-by-element definition of inclusion.
Hint 2
Each element may be included or omitted.
Worked solution

  1. Apply the element-by-element definition of inclusion.

  2. Calculate or simplify this relation.

    23=82^{3}=8

  3. The empty set and the full set both count.

The requested value is 8.

Checks and common pitfalls: The empty set and the full set both count.

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

02 / Standard#Worked example

How many proper subsets does A have?

∣A∣=3|A|=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Apply the element-by-element definition of inclusion.
Hint 2
Remove only the full set.
Worked solution

  1. Apply the element-by-element definition of inclusion.

  2. Calculate or simplify this relation.

    23−1=72^{3}-1=7

  3. For a nonempty set, the empty set remains a proper subset.

The requested value is 7.

Checks and common pitfalls: For a nonempty set, the empty set remains a proper subset.

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

03 / Transfer#Worked example

Count the nonempty subsets that omit one specified element.

∣A∣=3|A|=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Apply the element-by-element definition of inclusion.
Hint 2
First omit the element, then exclude the empty choice.
Worked solution

  1. Apply the element-by-element definition of inclusion.

  2. Calculate or simplify this relation.

    22−1=32^{2}-1=3

  3. The two restrictions must both be applied.

The requested value is 3.

Checks and common pitfalls: The two restrictions must both be applied.

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

04 / Foundation#Your turn

How many subsets does A have?

∣A∣=4|A|=4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Apply the element-by-element definition of inclusion.
Hint 2
Each element may be included or omitted.
Worked solution

  1. Apply the element-by-element definition of inclusion.

  2. Calculate or simplify this relation.

    24=162^{4}=16

  3. The empty set and the full set both count.

The requested value is 16.

Checks and common pitfalls: The empty set and the full set both count.

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

05 / Foundation#Your turn

How many proper subsets does A have?

∣A∣=4|A|=4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Apply the element-by-element definition of inclusion.
Hint 2
Remove only the full set.
Worked solution

  1. Apply the element-by-element definition of inclusion.

  2. Calculate or simplify this relation.

    24−1=152^{4}-1=15

  3. For a nonempty set, the empty set remains a proper subset.

The requested value is 15.

Checks and common pitfalls: For a nonempty set, the empty set remains a proper subset.

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

06 / Foundation#Your turn

How many subsets must contain one specified element?

∣A∣=4|A|=4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Apply the element-by-element definition of inclusion.
Hint 2
Fix that element; the others remain free.
Worked solution

  1. Apply the element-by-element definition of inclusion.

  2. Calculate or simplify this relation.

    23=82^{3}=8

  3. A required element contributes no extra binary choice.

The requested value is 8.

Checks and common pitfalls: A required element contributes no extra binary choice.

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

07 / Foundation#Your turn

Explain why these statements are equivalent.

0∈A;{0}⊆A0\in A;\quad \{0\}\subseteq A
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Apply the element-by-element definition of inclusion.
Hint 2
A singleton has one element to check.
Worked solution

  1. Apply the element-by-element definition of inclusion.

  2. Calculate or simplify this relation.

    {0}⊆A  ⟺  (∀x∈{0},x∈A)  ⟺  0∈A\{0\}\subseteq A\iff(\forall x\in\{0\},x\in A)\iff0\in A

  3. Membership and inclusion use different types of objects.

Both say that zero belongs to A.

Checks and common pitfalls: Membership and inclusion use different types of objects.

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

Prove that the empty set is a subset of every set.

∅⊆A\varnothing\subseteq A
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Apply the element-by-element definition of inclusion.
Hint 2
A counterexample would have to belong to the empty set.
Worked solution

  1. Apply the element-by-element definition of inclusion.

  2. Calculate or simplify this relation.

    ¬(∅⊆A)⇒∃x∈∅:x∉A\neg(\varnothing\subseteq A)\Rightarrow\exists x\in\varnothing:x\notin A

  3. The proposed counterexample cannot exist.

No element of the empty set can violate inclusion.

Checks and common pitfalls: The proposed counterexample cannot exist.

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.

09 / Standard#Your turn

Prove transitivity of inclusion.

A⊆B, B⊆CA\subseteq B,\ B\subseteq C
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Apply the element-by-element definition of inclusion.
Hint 2
Follow an arbitrary element through the two inclusions.
Worked solution

  1. Apply the element-by-element definition of inclusion.

  2. Calculate or simplify this relation.

    x∈A⇒x∈B⇒x∈Cx\in A\Rightarrow x\in B\Rightarrow x\in C

  3. An arbitrary-element argument proves the inclusion for the whole set.

Every member of A is a member of C.

Checks and common pitfalls: An arbitrary-element argument proves the inclusion for the whole 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.

10 / Standard#Your turn

Count the nonempty subsets that omit one specified element.

∣A∣=4|A|=4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Apply the element-by-element definition of inclusion.
Hint 2
First omit the element, then exclude the empty choice.
Worked solution

  1. Apply the element-by-element definition of inclusion.

  2. Calculate or simplify this relation.

    23−1=72^{3}-1=7

  3. The two restrictions must both be applied.

The requested value is 7.

Checks and common pitfalls: The two restrictions must both be applied.

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

11 / Standard#Your turn

How many subsets does A have?

∣A∣=5|A|=5
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Apply the element-by-element definition of inclusion.
Hint 2
Each element may be included or omitted.
Worked solution

  1. Apply the element-by-element definition of inclusion.

  2. Calculate or simplify this relation.

    25=322^{5}=32

  3. The empty set and the full set both count.

The requested value is 32.

Checks and common pitfalls: The empty set and the full set both count.

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

12 / Transfer#Your turn

How many proper subsets does A have?

∣A∣=5|A|=5
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Apply the element-by-element definition of inclusion.
Hint 2
Remove only the full set.
Worked solution

  1. Apply the element-by-element definition of inclusion.

  2. Calculate or simplify this relation.

    25−1=312^{5}-1=31

  3. For a nonempty set, the empty set remains a proper subset.

The requested value is 31.

Checks and common pitfalls: For a nonempty set, the empty set remains a proper subset.

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

13 / Transfer#Your turn

How many subsets must contain one specified element?

∣A∣=5|A|=5
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Apply the element-by-element definition of inclusion.
Hint 2
Fix that element; the others remain free.
Worked solution

  1. Apply the element-by-element definition of inclusion.

  2. Calculate or simplify this relation.

    24=162^{4}=16

  3. A required element contributes no extra binary choice.

The requested value is 16.

Checks and common pitfalls: A required element contributes no extra binary choice.

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

Focus on one question

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

Enable JavaScript for a summary of your local work.

    Choose a foundation skill to revisit ↗

    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for basic relations between sets?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Apply basic relations between sets with explicit conditions.
    • Inclusion: Every member of A must belong to B.
      A⊆BA\subseteq B
    • Equality: Equality requires inclusion in both directions.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Every member of A must belong to B.
    • Expected reasoning: Equality requires inclusion in both directions.
    • Expected correction: Distinguish an element from its singleton set.

    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 ↗