LEARN · EXPLAIN · REVISE

Classify being an integer as a condition for being rational.

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

高一必修第一册(A版).pdf · 1.4 · PDF 24 / printed page 17

TOPIC 01

Sufficient and necessary conditions

Build understanding of sufficient and necessary conditions through definitions, contrasting cases and justified applications.

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

01 / Transfer#Your turn

x\in\mathbb Z;\quad x\in\mathbb Q

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the definition of a rational number.

Hint 2

Find a rational number that is not an integer.

Worked solution

Use the definition of a rational number.
Calculate or simplify this relation.
$n=n/1\in\mathbb Q;\quad1/2\in\mathbb Q\setminus\mathbb Z$
The strict inclusion of number sets determines the implication directions.

Sufficient but not necessary.

Checks and common pitfalls: The strict inclusion of number sets determines the implication directions.

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 sufficient and necessary conditions?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Apply sufficient and necessary conditions with explicit conditions.
Implication: If p implies q, p is sufficient for q and q is necessary for p.
$p\Rightarrow q$
Equivalence: Both directions are required for a necessary and sufficient condition.
$p\iff q$
Close with: conditions → representation → reasoning → check.

Expected reasoning: If p implies q, p is sufficient for q and q is necessary for p.
Expected reasoning: Both directions are required for a necessary and sufficient condition.
Expected correction: A converse is not automatic.