LEARN · EXPLAIN · REVISE

Basic operations on sets

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

高一必修第一册(A版).pdf · 1.3 · PDF 17 / printed page 10

Revisit first: Basic relations between sets

TOPIC 01

Basic operations on sets

Build understanding of basic operations on sets through definitions, contrasting cases and justified applications.

What you will be able to explain

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

And / or

Intersection uses both conditions; union uses at least one.

A\cap B,\quad A\cup B

Complement

A complement is relative to a specified universal set.

U\setminus A

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 operations on 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.

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

Find the cardinality of the intersection.

A=\{1,\ldots,4\},\ B=\{2,\ldots,6\}

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate membership into “and”, “or”, or “not”.

Hint 2

Keep members occurring in both sets.

Worked solution

Translate membership into “and”, “or”, or “not”.
Calculate or simplify this relation.
$A\cap B=\{2,3,4\}$
Intersection imposes both conditions.

The requested value is 3.

Checks and common pitfalls: Intersection imposes both conditions.

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

02 / Standard#Worked example

Find the cardinality of the union.

|A|=6,\ |B|=5,\ |A\cap B|=2

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate membership into “and”, “or”, or “not”.

Hint 2

Remove the overlap counted twice.

Worked solution

Translate membership into “and”, “or”, or “not”.
Calculate or simplify this relation.
$|A\cup B|=6+5-2=9$
Every union member must be counted once.

The requested value is 9.

Checks and common pitfalls: Every union member must be counted once.

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

03 / Transfer#Worked example

How many elements belong to exactly one of A and B?

|A|=7,\ |B|=6,\ |A\cap B|=3

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate membership into “and”, “or”, or “not”.

Hint 2

Delete the intersection from each set.

Worked solution

Translate membership into “and”, “or”, or “not”.
Calculate or simplify this relation.
$(|A|-3)+(|B|-3)=7$
Exactly one is different from at least one.

The requested value is 7.

Checks and common pitfalls: Exactly one is different from at least one.

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

04 / Foundation#Your turn

Find the cardinality of the intersection.

A=\{1,\ldots,5\},\ B=\{3,\ldots,8\}

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate membership into “and”, “or”, or “not”.

Hint 2

Keep members occurring in both sets.

Worked solution

Translate membership into “and”, “or”, or “not”.
Calculate or simplify this relation.
$A\cap B=\{3,4,5\}$
Intersection imposes both conditions.

The requested value is 3.

Checks and common pitfalls: Intersection imposes both conditions.

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

05 / Foundation#Your turn

Find the cardinality of the union.

|A|=7,\ |B|=6,\ |A\cap B|=2

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate membership into “and”, “or”, or “not”.

Hint 2

Remove the overlap counted twice.

Worked solution

Translate membership into “and”, “or”, or “not”.
Calculate or simplify this relation.
$|A\cup B|=7+6-2=11$
Every union member must be counted once.

The requested value is 11.

Checks and common pitfalls: Every union member must be counted once.

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

06 / Foundation#Your turn

Find the cardinality of the complement.

A\subseteq U,\ |U|=14,\ |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

Translate membership into “and”, “or”, or “not”.

Hint 2

Partition the universal set into two disjoint parts.

Worked solution

Translate membership into “and”, “or”, or “not”.
Calculate or simplify this relation.
$|U\setminus A|=14-5=9$
The complement depends on the specified universal set.

The requested value is 9.

Checks and common pitfalls: The complement depends on the specified universal set.

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

07 / Foundation#Your turn

Prove the complement-of-union rule.

U\setminus(A\cup B)

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate membership into “and”, “or”, or “not”.

Hint 2

Negate “in A or in B”.

Worked solution

Translate membership into “and”, “or”, or “not”.
Calculate or simplify this relation.
$x\notin A\cup B\iff(x\notin A)\land(x\notin B)$
Negating “or” produces “and”.

It equals (U∖A)∩(U∖B).

Checks and common pitfalls: Negating “or” produces “and”.

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

A group has 22 people; 9 take music, 10 sport, and 3 both. How many take neither?

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate membership into “and”, “or”, or “not”.

Hint 2

First find the union, then its complement.

Worked solution

Translate membership into “and”, “or”, or “not”.
Calculate or simplify this relation.
$22-(9+10-3)=6$
Neither means outside the union.

The requested value is 6.

Checks and common pitfalls: Neither means outside the union.

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

09 / Standard#Your turn

Find the interval intersection.

[-3,6)\cap(3,10]

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate membership into “and”, “or”, or “not”.

Hint 2

Use the tighter bound at each end.

Worked solution

Translate membership into “and”, “or”, or “not”.
Calculate or simplify this relation.
$3<x<6$
Each endpoint fails at least one original condition.

The intersection is (3,6).

Checks and common pitfalls: Each endpoint fails at least one original condition.

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 belong to exactly one of A and B?

|A|=8,\ |B|=7,\ |A\cap B|=3

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate membership into “and”, “or”, or “not”.

Hint 2

Delete the intersection from each set.

Worked solution

Translate membership into “and”, “or”, or “not”.
Calculate or simplify this relation.
$(|A|-3)+(|B|-3)=9$
Exactly one is different from at least one.

The requested value is 9.

Checks and common pitfalls: Exactly one is different from at least one.

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

11 / Standard#Your turn

Find the cardinality of the intersection.

A=\{1,\ldots,6\},\ B=\{4,\ldots,10\}

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate membership into “and”, “or”, or “not”.

Hint 2

Keep members occurring in both sets.

Worked solution

Translate membership into “and”, “or”, or “not”.
Calculate or simplify this relation.
$A\cap B=\{4,5,6\}$
Intersection imposes both conditions.

The requested value is 3.

Checks and common pitfalls: Intersection imposes both conditions.

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

12 / Transfer#Your turn

Find the cardinality of the union.

|A|=8,\ |B|=7,\ |A\cap B|=2

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate membership into “and”, “or”, or “not”.

Hint 2

Remove the overlap counted twice.

Worked solution

Translate membership into “and”, “or”, or “not”.
Calculate or simplify this relation.
$|A\cup B|=8+7-2=13$
Every union member must be counted once.

The requested value is 13.

Checks and common pitfalls: Every union member must be counted once.

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

13 / Transfer#Your turn

Find the cardinality of the complement.

A\subseteq U,\ |U|=17,\ |A|=6

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate membership into “and”, “or”, or “not”.

Hint 2

Partition the universal set into two disjoint parts.

Worked solution

Translate membership into “and”, “or”, or “not”.
Calculate or simplify this relation.
$|U\setminus A|=17-6=11$
The complement depends on the specified universal set.

The requested value is 11.

Checks and common pitfalls: The complement depends on the specified universal set.

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

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.

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Teacher preparation and assessment

Question sequence

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

Board plan

Apply basic operations on sets with explicit conditions.
And / or: Intersection uses both conditions; union uses at least one.
$A\cap B,\quad A\cup B$
Complement: A complement is relative to a specified universal set.
$U\setminus A$
Close with: conditions → representation → reasoning → check.

Anticipated thinking

Expected reasoning: Intersection uses both conditions; union uses at least one.
Expected reasoning: A complement is relative to a specified universal set.
Expected correction: Do not count the overlap twice.

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.