← Senior Mathematics Studio

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

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

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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.

Try it. Leave your reasoning visible.

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

01 / Standard#Your turn

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

\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

Apply the element-by-element definition of inclusion.
Calculate or simplify this relation.
$\neg(\varnothing\subseteq A)\Rightarrow\exists x\in\varnothing:x\notin A$
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.

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.
$A\subseteq B$
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.