Sec 4 · G3 (Express)

Sets

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1. Set notation

Symbol	Meaning	Example
$\xi$	universal set	$\xi = \{1, 2, 3, \ldots, 10\}$
$\{\ \}$	the set of	$A = \{3, 6, 9\}$
$\{x : \ldots\}$	the set of all $x$ such that …	$A = \{x : x \in \xi, \ x \text{ is a multiple of } 3\}$
$\in$	is an element of	$6 \in A$
$\notin$	is not an element of	$5 \notin A$
$n(A)$	number of elements in $A$	$n(A) = 3$
$A'$	complement: in $\xi$ but not in $A$	$A' = \{1, 2, 4, 5, 7, 8, 10\}$
$\varnothing$ or $\{\ \}$	empty set	$\{x : x \in \xi, \ x > 10\} = \varnothing$

📖 Worked example

$\xi = \{x : x \text{ is an integer and } 1 \le x \le 10\}$ and $A$ is the set of multiples of 3 in $\xi$ . List the elements of $A'$ .

Step 1 of 2

List the multiples of 3 from 1 to 10

A = \{3, 6, 9\}

Integers	How many
from $3$ to $10$	$10 - 3 + 1 = 8$
from $2$ to $k$	$k - 2 + 1 = k - 1$

📖 Worked example

$\xi = \{x : x \text{ is an integer and } 8 \le x \le k\}$ and $A$ is the set of multiples of 3 in $\xi$ . Given that $n(A) = 3$ , find the greatest possible value of $k$ .

Step 1 of 3

The multiples of 3 from 8 upwards

9, 12, 15, \ldots

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2. Union, intersection and Venn diagrams

Symbol	Meaning	Example ( $B$ = even numbers)
$A \cap B$	intersection: in $A$ and in $B$	$\{6\}$
$A \cup B$	union: in $A$ or $B$ or both	$\{2, 3, 4, 6, 8, 9, 10\}$
$A \cap B'$	in $A$ but not in $B$	$\{3, 9\}$
$(A \cup B)'$	in neither	$\{1, 5, 7\}$

Six Venn diagrams with the region shaded: A ∩ B is the overlap; A ∪ B is both circles; A′ is everything outside A; A ∩ B′ is A without the overlap; A′ ∩ B is B without the overlap; (A ∪ B)′ is outside both circles.

Number in the union

\displaystyle n(A \cup B) = n(A) + n(B) - n(A \cap B)

$n(A) = 12$ , $n(B) = 20$ , $n(A \cap B) = 7$ : $n(A \cup B) = 25$

📖 Worked example

The Venn diagram shows the elements of $\xi$ , $A$ and $B$ . List the elements of $A \cap B'$ .

Step 1 of 2

Which regions make up the set? Yes or No for each

A \cap B'

: in A but not in B.Regions: A only.

📖 Worked example

$n(A) = 12$ , $n(B) = 20$ and $n(A \cap B) = 7$ . Find $n(A \cup B)$ .

Step 1 of 1

Use n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

\mathstrut n(A \cup B) = 12 + 20 - 7

n(A \cup B) = 25

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3. Subsets

Symbol	Meaning	Example
$\subseteq$	is a subset of (may be the whole set)	$\{3, 6\} \subseteq A$
$\subset$	is a proper subset of (not the whole set)	$\{3, 6\} \subset A$
$\not\subseteq$	is not a subset of	$\{3, 4\} \not\subseteq A$

📖 Worked example

$A = \{1, 3, 5, 7\}$ . State whether $\varnothing \subset A$ is true or false.

Step 1 of 2

∈ compares an element with a set; ⊆ and ⊂ compare two sets

\varnothing

is a subset of

A

, and it leaves out the elements of

A

, so it is a proper subset.

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4. Problems with Venn diagrams

30 people: 21 cat, 20 dog, 3 neither. x own both

\displaystyle (21 - x) + x + (20 - x) + 3 = 30

e.g.

\displaystyle (21 - x) + x + (20 - x) + 3 = 30

\displaystyle 44 - x = 30

\displaystyle x = 14

Least and greatest overlap

\displaystyle n(A) + n(B) - n(\xi)

least: $21 + 20 - 30 = 11$ (if this is $0$ or less, the least is $0$ )
greatest: the smaller group, $20$

📖 Worked example

In a group of $30$ people, $21$ own a cat and $20$ own a dog. $3$ do neither. Find the number of people who do both.

Step 1 of 2

Let x be the number who do both. Write each 'only' region in terms of x

\text{own a cat only} = 21 - x

\text{own a dog only} = 20 - x

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Sec 4 · G3 (Express)

Sets

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1. Set notation

Symbol	Meaning	Example
$\xi$	universal set	$\xi = \{1, 2, 3, \ldots, 10\}$
$\{\ \}$	the set of	$A = \{3, 6, 9\}$
$\{x : \ldots\}$	the set of all $x$ such that …	$A = \{x : x \in \xi, \ x \text{ is a multiple of } 3\}$
$\in$	is an element of	$6 \in A$
$\notin$	is not an element of	$5 \notin A$
$n(A)$	number of elements in $A$	$n(A) = 3$
$A'$	complement: in $\xi$ but not in $A$	$A' = \{1, 2, 4, 5, 7, 8, 10\}$
$\varnothing$ or $\{\ \}$	empty set	$\{x : x \in \xi, \ x > 10\} = \varnothing$

📖 Worked example

$\xi = \{x : x \text{ is an integer and } 1 \le x \le 10\}$ and $A$ is the set of multiples of 3 in $\xi$ . List the elements of $A'$ .

Step 1 of 2

List the multiples of 3 from 1 to 10

A = \{3, 6, 9\}

Integers	How many
from $3$ to $10$	$10 - 3 + 1 = 8$
from $2$ to $k$	$k - 2 + 1 = k - 1$

📖 Worked example

$\xi = \{x : x \text{ is an integer and } 8 \le x \le k\}$ and $A$ is the set of multiples of 3 in $\xi$ . Given that $n(A) = 3$ , find the greatest possible value of $k$ .

Step 1 of 3

The multiples of 3 from 8 upwards

9, 12, 15, \ldots

Practise this →

2. Union, intersection and Venn diagrams

Symbol	Meaning	Example ( $B$ = even numbers)
$A \cap B$	intersection: in $A$ and in $B$	$\{6\}$
$A \cup B$	union: in $A$ or $B$ or both	$\{2, 3, 4, 6, 8, 9, 10\}$
$A \cap B'$	in $A$ but not in $B$	$\{3, 9\}$
$(A \cup B)'$	in neither	$\{1, 5, 7\}$

Number in the union

\displaystyle n(A \cup B) = n(A) + n(B) - n(A \cap B)

$n(A) = 12$ , $n(B) = 20$ , $n(A \cap B) = 7$ : $n(A \cup B) = 25$

📖 Worked example

The Venn diagram shows the elements of $\xi$ , $A$ and $B$ . List the elements of $A \cap B'$ .

Step 1 of 2

Which regions make up the set? Yes or No for each

A \cap B'

: in A but not in B.Regions: A only.

📖 Worked example

$n(A) = 12$ , $n(B) = 20$ and $n(A \cap B) = 7$ . Find $n(A \cup B)$ .

Step 1 of 1

Use n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

\mathstrut n(A \cup B) = 12 + 20 - 7

n(A \cup B) = 25

Practise this →

3. Subsets

Symbol	Meaning	Example
$\subseteq$	is a subset of (may be the whole set)	$\{3, 6\} \subseteq A$
$\subset$	is a proper subset of (not the whole set)	$\{3, 6\} \subset A$
$\not\subseteq$	is not a subset of	$\{3, 4\} \not\subseteq A$

📖 Worked example

$A = \{1, 3, 5, 7\}$ . State whether $\varnothing \subset A$ is true or false.

Step 1 of 2

∈ compares an element with a set; ⊆ and ⊂ compare two sets

\varnothing

is a subset of

A

, and it leaves out the elements of

A

, so it is a proper subset.

Practise this →

4. Problems with Venn diagrams

30 people: 21 cat, 20 dog, 3 neither. x own both

\displaystyle (21 - x) + x + (20 - x) + 3 = 30

e.g.

\displaystyle (21 - x) + x + (20 - x) + 3 = 30

\displaystyle 44 - x = 30

\displaystyle x = 14

Least and greatest overlap

\displaystyle n(A) + n(B) - n(\xi)

least: $21 + 20 - 30 = 11$ (if this is $0$ or less, the least is $0$ )
greatest: the smaller group, $20$

📖 Worked example

In a group of $30$ people, $21$ own a cat and $20$ own a dog. $3$ do neither. Find the number of people who do both.

Step 1 of 2

Let x be the number who do both. Write each 'only' region in terms of x

\text{own a cat only} = 21 - x

\text{own a dog only} = 20 - x

Practise this →

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