Speak Maths · Last school years

Sets: who is on the shelf

A set is a collection. A few signs say who belongs to it, what two collections share, and what is left over.

Draft: written by an AI helper and not yet checked by a teacher.

The alphabet

Read each sign aloud, the way it is written under it.

$x \in A x \in / A$ Say: x is an element of A; x is not an element of AMeans: x is on shelf A; the slash says it is not
$A \subseteq B$ Say: A is a subset of BMeans: every element of A is also in B
$A \cup B$ Say: A union BMeans: everything that is in A, in B, or in both
$A \cap B$ Say: A intersect BMeans: only what is in both A and B
$A ∖ B$ Say: A minus BMeans: what is in A and not in B
$\emptyset$ Say: the empty setMeans: the set with nothing in it
$N \subset Z \subset Q \subset R$ Say: the naturals sit inside the integers, inside the rationals, inside the realsMeans: number families, each inside the next: counting, whole, fractions, the whole line
$[2, 5) = {x \in R ∣ 2 \leq x < 5}$ Say: the interval from 2, included, to 5, not included; the set of all real x such that 2 is at most x and x is less than 5Means: a square bracket keeps the end, a round one leaves it out

The story: The library shelves

Subtitles:

Tap a line to see its words.

$A = {1, 2, 3, 4}$ Shelf A holds the books numbered 1, 2, 3 and 4.
$B = {3, 4, 5, 6}$ Shelf B holds the books numbered 3, 4, 5 and 6.
$2 \in A 2 \in / B$ Book 2 is on shelf A, and not on shelf B.
$A \cap B = {3, 4}$ Books on both shelves: 3 and 4.
$A \cup B = {1, 2, 3, 4, 5, 6}$ Together the two shelves hold books 1 to 6.
$A ∖ B = {1, 2}$ On shelf A but not on shelf B: books 1 and 2.
$C = {3, 4} \subseteq A \emptyset \subseteq A$ The pile on the reading table, books 3 and 4, came wholly from shelf A. The empty trolley fits inside A too.
${n \in N ∣ 1 \leq n \leq 6} = A \cup B$ The natural numbers from 1 to 6 are exactly the books on the two shelves.
$2.5 \in [2, 5) 5 \in / [2, 5)$ A slot takes books 2 to 5 centimetres thick: 2 allowed, 5 too thick. A 2.5 book fits; a 5 does not.
${1, 2, 3, 4, 5, 6, 7} ∖ (A \cup B) = {7}$ Take both shelves away from books 1 to 7, and only book 7 remains. It is lost.

In words this story is 621 letters long; in symbols, 95.

Read it

What does it say?

$B \subseteq A$

$A \cap B = \emptyset$

${x \in R ∣ x \geq 3}$

$2 \in R ∖ Q$

$7 \in / A$

$A ∖ B = \emptyset$

$(0, 1]$

$A \cup B = A$

$- 2 \in Z ∖ N$

${x \in Z ∣ - 1 \leq x < 2} = {- 1, 0, 1}$

$A \cap (B \cup C)$

$Q \subset R$

Write it

Which notation says this?

The elements that are in A or in B, or in both.

The real numbers from 1 to 4, with 1 included and 4 left out.

The set of all real numbers x with x greater than 2.

The elements that are in both A and B.

x is not an element of B.

The set of all integers x with x squared less than 10.

Every element of C is also an element of D.

The real numbers strictly between 0 and 1.

What is in B but not in A.

A and B share no element.

Three is a natural number.

The empty set is a subset of every set A.

Build it

Tap the tiles in order. One tile does not belong.

The overlap of A and B is a subset of A.

x is a real number that is not rational.

The real numbers from 0 to 3, with 3 included and 0 left out.

x is an element of A union B.

The set of natural numbers n such that n is less than 4.

The overlap of A and B is a subset of their union.

Challenges

Creative puzzles that use every sign of this chapter. Take your time; there is no clock.

1 Three toy boxes

Five toys and three boxes. Judge each sentence about the boxes: true or false?

	$A$	$B$	$C$
$1$	✓
$2$	✓	✓	✓
$3$	✓	✓	✓
$4$		✓
$5$

A tick means the toy is in the box: row 2 is ticked under A, B and C, so 2 is in all three boxes.

$C \subseteq A \cap B$
$A \subseteq B$
$B ∖ A = {4}$
$(A ∖ B) \cap C = \emptyset$
$5 \in A \cup B$
$1 \in / B$

2 Which family?

Pair each line with what it says.

Tap a line, then its meaning.

3 Shelf talk

Match the notation with what it says about shelves A and B.

Tap a line, then its meaning.

4 A bracket slip

This should say: the interval from 2 (kept) to 5 (left out) is the set of real numbers x from 2 up to, but not including, 5. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

5 Only in A

This should say: x is in A without B exactly when x is in A and x is not in B. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

6 The missing shelf

Four books in all. Shelf A has books 1 and 4. The shelves share no book, and together they hold all four. What is on shelf B?

$A = {1, 4}, A \cap B = \emptyset, A \cup B = {1, 2, 3, 4} B = ?$

7 Squares under 30

Which set is this?

${n \in N ∣ n^{2} < 30}$

8 Even or odd

Let E be the even whole numbers and O the odd ones. Say in symbols: no whole number is both, and every whole number is one of the two. Then compare.

Your version

9 Keys on rings

Four keys, three rings; a ring is the set of keys hanging on it. Judge each sentence: true or false?

	$A$	$B$	$C$
$k_{1}$	✓	✓
$k_{2}$	✓
$k_{3}$		✓	✓
$k_{4}$			✓

A tick means the key hangs on that ring.

$A \cap B = {k_{1}}$
$A \cap C = \emptyset$
$B \subseteq A \cup C$
$k_{4} \in A \cup B$
$C ∖ B = {k_{3}}$
$A \subseteq B \cup C$

10 Four numbers, four families

Four numbers and the four number families. Judge each sentence: true or false?

	$N$	$Z$	$Q$	$R$
$- 2$		✓	✓	✓
$\frac{1}{2}$			✓	✓
$2$				✓
$3$	✓	✓	✓	✓

A tick means the number belongs to that family.

$2 \in / Q$
$- 2 \in N$
$\frac{1}{2} \in Q ∖ Z$
${- 2, 3} \subseteq Z$
$R \subseteq Q$
$2 \in Z \cup Q$

11 Brackets

Five ways to write 0 and 1 side by side. Pair each with what it says.

Tap a line, then its meaning.

12 Two circles

Picture two overlapping circles A and B. Pair each sign with the region it names.

Tap a line, then its meaning.

13 Nothing new

This should say: if A is a subset of B, then A together with B is just B. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

14 The number ladder

This should say: 3 is a natural number, and the naturals sit inside the reals, therefore 3 is real. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

15 The lost-and-found box

Box A alone holds item 1; the two boxes share items 2 and 3; box B holds nothing that A lacks. What is in box B?

$A ∖ B = {1}, A \cap B = {2, 3}, B ∖ A = \emptyset B = ?$

16 Rain or sun

Let D be the days of one week, R the rainy days and S the sunny days. Say in symbols: every day was rainy or sunny, and at least one day was both. Then compare.

Your version

Tell your own

Tell a three-line story about two boxes of toys, in signs only: what is in both, what is in one but not the other. Then read it aloud.

Kept on this device only.

Sets: who is on the shelf

The alphabet

The story: The library shelves

Read it

Write it

Build it

Challenges

1 Three toy boxes

2 Which family?

3 Shelf talk

4 A bracket slip

5 Only in A

6 The missing shelf

7 Squares under 30

8 Even or odd

9 Keys on rings

10 Four numbers, four families

11 Brackets

12 Two circles

13 Nothing new

14 The number ladder

15 The lost-and-found box

16 Rain or sun

Tell your own

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