International Baccalaureate · DP 1: Applications and Interpretation HL

Sets and Venn Diagrams

⌚ ~170 min ▢ 19 blocks ✓ 77 exercises

I Sets

I.1 Sets and Elements

Recappreviously studied — click to reread

Definition — Set

A set is a collection of objects, called elements.
We list its elements between curly brackets, each element only once and in any order: $\{3, 8, 9\} = \{9, 3, 8\}$.

Definition — Element

An element is an object contained in a set.
$\in$ means “is an element of” or “belongs to”.
$\notin$ means “is not an element of” or “does not belong to”.

Example

$2\in \{1,2,3,4,5,6\}$ and $7\notin \{1,2,3,4,5,6\}$.

Definition — Equal sets

Two sets are equal if they have exactly the same elements.

Example

Determine if the sets $\{2,6,4\}$ and $\{2,4,6\}$ are equal.

Answer

Yes, the sets $\{2,6,4\}$ and $\{2,4,6\}$ are equal because they contain the same elements: $2$, $4$, and $6$.

Definition — Empty Set

The empty set is a set with no elements. It is written as $\{\}$ or $\emptyset$.

Skills to practice

Checking Membership
Checking Set Equality

I.2 Subsets

Recappreviously studied — click to reread

Definition — Subset

A set $A$ is a subset of a set $B$ if every element in $A$ is also in $B$. We write this as $A \subseteq B$. Some books write $A \subset B$, as in the nesting of the number sets.

If at least one element of $A$ is not in $B$, then $A$ is not a subset of $B$, and we write $A \not\subseteq B$.
Every set is a subset of itself: $A \subseteq A$.
The empty set is a subset of every set: it has no element outside $A$, so $\emptyset \subseteq A$.

Example

Let $B = \{1, 2, 3, 4, 5, 6\}$, $A = \{2, 4, 6\}$ and $C = \{2, 7\}$. Is $A \subseteq B$? Is $C \subseteq B$?

Answer

Check each element: $2$, $4$, and $6$ from $A$ are all in $B = \{1, 2, 3, 4, 5, 6\}$. Since every element of $A$ is in $B$, $A \subseteq B$.
In $C$, the element $2$ is in $B$, but the element $7$ is not. One element outside $B$ is enough: $C \not\subseteq B$.

Skills to practice

Checking Subsets

I.3 Set-Builder Notation

Recappreviously studied — click to reread

Definition — Set-Builder Notation

The set-builder notation is a way to describe a set by giving a rule that its elements must follow. It is written like this:$$\{x \in E \mid \text{condition on } x\}$$ In this notation, $x$ represents an element from a set $E$, and the symbol $\mid$ (or sometimes $:$) separates $x$ from the condition it must meet.
It reads: “the set of all $x$ in $E$ such that $x$ satisfies the condition.”

Example

Let $E=\{1, 2, 3, 4, 5, 6\}$ be the set of results from rolling a standard six-sided die

.
What are the elements of the set $\{x \in E \mid x \text{ is even}\}$?

Answer

The set $\{x \in E \mid x \text{ is even}\}$ contains all the even numbers in $E$. Given $E = \{1, 2, 3, 4, 5, 6\}$, the even numbers are $2$, $4$, and $6$, so:$$\{x \in E \mid x \text{ is even}\} = \{2, 4, 6\}$$

Skills to practice

Listing the Elements of a Set in Set-Builder Notation

I.4 The Number of Elements of a Set

Recappreviously studied — click to reread

Definition — Cardinality

$\Card{A}$ denotes the number of elements in the set $A$.

Definition — Finite and Infinite Sets

A finite set has a finite number of elements. That is, its elements can all be counted, and the counting comes to an end.
An infinite set has an unlimited number of elements; it is not finite.

Example

$\{1, 2, 3\}$ is finite because it has exactly 3 elements.
$\N = \{0, 1, 2, 3, \dots\}$ is infinite because there is no end to counting.

Skills to practice

Counting the Elements of a Set

I.5 Intersection and Union

Recappreviously studied — click to reread

Definition — Intersection

The intersection of two sets $A$ and $B$, written $A \cap B$, is the set of elements that are in both $A$ and $B$.

Example

What is the intersection $\{1, 2, 3\} \cap \{2, 3, 4\}$?

Answer

For the intersection $\cap$, include all common elements: $\textcolor{colorprop}{2}$ and $\textcolor{olive}{3}$. So $$\{\textcolor{colordef}{1}, \textcolor{colorprop}{2}, \textcolor{olive}{3}\} \cap \{ \textcolor{colorprop}{2}, \textcolor{olive}{3},\textcolor{brown}{4}\} = \{\textcolor{colorprop}{2},\textcolor{olive}{3}\}$$

Definition — Union

The union of two sets $A$ and $B$, written $A \cup B$, is the set of all elements in $A$ or $B$ (or both).

Example

What is the union $\{1, 2, 3\} \cup \{2, 3, 4\}$?

Answer

For the union $\cup$, include all elements from both sets without repeats: $\textcolor{colordef}{1}, \textcolor{colorprop}{2}, \textcolor{olive}{3},\textcolor{brown}{4}$. So, $$\{\textcolor{colordef}{1}, \textcolor{colorprop}{2}, \textcolor{olive}{3}\} \cup \{ \textcolor{colorprop}{2}, \textcolor{olive}{3},\textcolor{brown}{4}\} = \{\textcolor{colordef}{1}, \textcolor{colorprop}{2}, \textcolor{olive}{3},\textcolor{brown}{4}\}$$

Skills to practice

Intersection and Union

I.6 The Complement of a Set

Recappreviously studied — click to reread

Definition — Universal set

A universal set, written $U$, is the set of all elements considered. For example, for the possible results of rolling a die, $U = \{1, 2, 3, 4, 5, 6\}$.

Definition — Complement

The complement of a set $A$, denoted $A'$ (also written $\overline{A}$ or $U \setminus A$), consists of all elements in universal set $U$ that are not in $A$. Sets $A$ and $A'$ are said to be complementary.

Example

Given the universal set $U = \{1, 2, 3, 4, 5, 6\}$ and the set $A = \{1, 3, 5\}$, find the complement $A'$.

Answer

Start with the universal set $U = \{1, 2, 3, 4, 5, 6\}$.
The set $A = \{1, 3, 5\}$ includes 1, 3, and 5.
The complement $A'$ is all the elements in $U$ that are not in $A$:
$$A' = \{2, 4, 6\}$$

Skills to practice

Finding the Complement

II Number Sets and Intervals

II.1 Special Number Sets

Recappreviously studied — click to reread

Number sets are collections of numbers defined by shared properties and they form a foundation of mathematics. We begin with the natural numbers ($\N$), used for counting. By including negative numbers, we obtain the integers ($\Z$). Allowing quotients of integers (with nonzero denominator) leads to the rational numbers ($\Q$). Among real numbers ($\R$), we also find irrational numbers (such as $\sqrt{2}$ and $\pi$), which together with the rationals fill the entire number line. Each of these sets contains the previous ones.

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Definition — Number Sets

The natural numbers, denoted $\N$, are the counting numbers, starting from zero:$$\N = \{0, 1, 2, 3, 4, 5, 6, \dots\}$$
The integers, denoted $\Z$, are the natural numbers and their opposites (negative, zero, positive):$$\Z = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$$
The terminating decimals (the set $\mathbb{D}$ of French textbooks) are numbers that can be written as a fraction where the denominator is a power of 10:$$\mathbb{D} = \left\{ \frac{a}{10^n} \mid a \in \Z,\, n \in \N \right\}$$
The rational numbers, denoted $\Q$, are all numbers that can be written as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$:$$\Q = \left\{ \frac{p}{q} \mid p, q \in \Z,\, q \neq 0 \right\}$$
The real numbers, denoted $\R$, include all points on the number line. A real number that is not rational is called irrational, such as $\sqrt{2}$ and $\pi$: the real numbers are the rational numbers and the irrational numbers.

Proposition — Irrationality of $\sqrt{2}$

$\sqrt{2}$ is an irrational number.

Proof

Assume $\sqrt{2}$ is rational. Then $\sqrt{2} = \frac{p}{q}$ where $\frac{p}{q}$ is a fraction in lowest terms.
Squaring both sides: $2 = \frac{p^2}{q^2} \implies p^2 = 2q^2$.
This implies $p^2$ is even, so $p$ must be even — indeed, if $p$ were odd, its square would be odd. Let $p = 2k$.
Substitute back: $(2k)^2 = 2q^2 \implies 4k^2 = 2q^2 \implies 2k^2 = q^2$.
This implies $q^2$ is even, so $q$ must be even. If both $p$ and $q$ are even, they have a common factor of 2, contradicting that the fraction is in lowest terms.
Thus, $\sqrt{2}$ cannot be rational.

Example

$1\in \N, 1\in \Z, 1\in \mathbb{D},1\in \Q, 1\in\R$
$-5 \in \Z$, $-5 \notin \N$
$0.25 = \frac{25}{100} = \frac{1}{4} \in \mathbb{D}$ (and also in $\Q$ and $\R$)
$\frac{1}{3} \in \Q$ but $\frac{1}{3} \notin \mathbb{D}$
$\sqrt{2} \in \R$ but $\sqrt{2} \notin \Q$ (irrational)

Definition — Positive Numbers

A plus sign written as a superscript keeps only the positive elements of a set of numbers, those greater than $0$:

$\Z^+$ $= \{1, 2, 3, \dots\}$, the set of the positive integers;
$\Q^+$ $= \{x \in \Q \mid x > 0\}$, the set of the positive rational numbers;
$\R^+$ $= \{x \in \R \mid x > 0\}$, the set of the positive real numbers.

Zero is neither positive nor negative, so $0 \notin \Z^+$, $0 \notin \Q^+$ and $0 \notin \R^+$. The positive integers are the natural numbers other than $0$.

Example

$7 \in \Z^+$ and $-7 \notin \Z^+$.
$\frac{2}{3} \in \Q^+$, but $-\frac{2}{3} \notin \Q^+$, and $\sqrt{2} \notin \Q^+$ since $\sqrt{2}$ is not rational.
$\sqrt{2} \in \R^+$ and $\pi \in \R^+$, but $0 \notin \R^+$.

Proposition — Relationships Between Number Sets

The number sets are nested as follows:$$\N \subset \Z \subset \mathbb{D} \subset \Q \subset \R$$ This means: every natural number is an integer, every integer is a decimal number, every decimal number is a rational number, and every rational number is a real number.

Skills to practice

Checking Membership in the Sets of Positive Numbers

II.2 Intervals

Recappreviously studied — click to reread

Definition — Interval

An interval is the set of all real numbers between two endpoints, which may or may not be included in the set, or the set of all real numbers beyond one endpoint (going on without end to $+\infty$ or to $-\infty$), or the set $\R$ of all real numbers.

Example

The set of all real numbers between $0$ and $1$, including $1$ but not $0$, is an interval. It is written as $\{x \in \R \mid 0 < x \leq 1\}$.

Method — Representing Intervals on a Number Line

Intervals are often shown on a number line using these conventions:

An open point (empty circle) or a round bracket means that the endpoint is not included.
A closed point (filled circle) or a square bracket means that the endpoint is included.
An arrow shows that the interval extends indefinitely (to $+\infty$ or $-\infty$).

Example

The number line representation of $\{x \in \R \mid 0 < x \leq 1\}$ is:

Definition — Interval Notation

Let $a$ and $b$ be two real numbers with $a < b$.

Interval Notation	Set-builder notation	Number line representation
$\CloseBracketLeft a, b \CloseBracketRight$	$\{x \in \R \mid a \leq x \leq b\}$
$\CloseBracketLeft a, b \OpenBracketRight$	$\{x \in \R \mid a \leq x < b\}$
$\OpenBracketLeft a, b \CloseBracketRight$	$\{x \in \R \mid a < x \leq b\}$
$\OpenBracketLeft a, b \OpenBracketRight$	$\{x \in \R \mid a < x < b\}$
$\CloseBracketLeft a, +\infty\OpenBracketRight $	$\{x \in \R \mid a \leq x\}$
$\OpenBracketLeft a, +\infty \OpenBracketRight $	$\{x \in \R \mid a < x\}$
$\OpenBracketLeft -\infty, a\CloseBracketRight$	$\{x \in \R \mid x \leq a\}$
$\OpenBracketLeft -\infty, a\OpenBracketRight$	$\{x \in \R \mid x < a\}$

Two Notations for an Excluded Endpoint

An endpoint that is not included is written in one of two ways: with a round bracket, as in $(a, b)$, or with a square bracket turned outwards, as in $\left]a, b\right[$. Both notations are in use, and they describe the same interval: $$(a, b) = \left]a, b\right[, \qquad [a, b) = \left[a, b\right[, \qquad (a, +\infty) = \left]a, +\infty\right[.$$

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Infinity Is Not a Number

The symbols $-\infty$ and $+\infty$ are not real numbers: they only indicate that the interval goes on without end. That is why the side of $\infty$ is always excluded, with a round bracket: one writes $[2, +\infty)$, never $[2, +\infty]$.

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Method — Union and Intersection of Two Intervals

To combine two intervals, draw them on the same number line, one above the other.

The intersection $I \cap J$ is the zone covered by both colourings — where the two intervals overlap. It can be empty.
The union $I \cup J$ is the zone covered by at least one colouring. If a gap remains between the two zones — they do not overlap and do not share an endpoint that one of them includes, as in $[0, 1) \cup (1, 2]$ — the union is not an interval and stays written with the symbol $\cup$.

At each end, check carefully whether the endpoint is included or excluded.

Example

Let $I = [-1, 4]$ and $J = (2, +\infty)$. Find $I \cap J$ and $I \cup J$.

Answer

On the number line, the two intervals overlap between $2$ and $4$.

$I \cap J = (2, 4]$: the endpoint $2$ is excluded ($2 \notin J$), the endpoint $4$ is included ($4 \in I$ and $4 \in J$).
$I \cup J = [-1, +\infty)$: the two zones together cover everything from $-1$ onwards, without a gap.

Skills to practice

Combining Intervals with Union and Intersection

III Venn Diagrams

III.1 Reading a Venn Diagram

Recappreviously studied — click to reread

Definition — Venn Diagram

A Venn diagram uses a rectangle to show the universal set $U$ and circles to represent other sets within it.

Example

Here is a Venn diagram for $U = \{1, 2, 3, 4, 5, 6\}$ and $A = \{1, 3, 5\}$:

Definition — Key Venn Diagram Concepts

This table shows common set operations and their Venn diagrams:

Notation	Meaning	Venn Diagram
$A$	Set $A$
$A'$	Complement of $A$ (everything in $U$ not in $A$)
$A \subseteq B$	$A$ is a subset of $B$
$A \cup B$	Union of $A$ and $B$ (all elements in $A$ or $B$)
$A \cap B$	Intersection of $A$ and $B$ (elements in both)
$A \cap B = \emptyset$	$A$ and $B$ are disjoint (no common elements)

Definition — Counting Elements

In a Venn diagram, we use round brackets around numbers to show how many elements are in each region.

Example

Consider this Venn diagram:

Here, there are 6 elements in $A\cap B$, 4 in $A$ but not in $B$, 5 in $B$ but not in $A$, and 1 outside both. Total: $A$ has $4 + 6 = 10$, $B$ has $6 + 5 = 11$.

Skills to practice

Identifying Sets on a Venn Diagram
Reading Numbers on a Venn Diagram

III.2 Counting with a Venn Diagram

Recappreviously studied — click to reread

Discover

In a class of $30$ pupils, $18$ play football, $15$ play basketball and $7$ play neither of these two sports.

Add the number of football players and the number of basketball players. Why is this total larger than the number of pupils who play at least one of the two sports?
How many pupils play both sports?

Answer

$18 + 15 = 33$, while only $30 - 7 = 23$ pupils play at least one of the two sports. A pupil who plays both sports is counted once among the football players and a second time among the basketball players: the sum counts these pupils twice.
The sum exceeds the number of players by exactly the number of pupils counted twice: $33 - 23 = 10$ pupils play both sports. The Venn diagram, with $F$ for football and $B$ for basketball, shows the four groups of pupils: Check: $8 + 10 + 5 + 7 = 30$ pupils.

Definition — Regions of a Venn Diagram

Two sets $A$ and $B$, drawn as two overlapping circles inside the universal set $U$, divide the diagram into four regions. Every element of $U$ lies in exactly one of them.

$A \cap B'$: the elements of $A$ that are not in $B$, “in $A$ only”.
$A \cap B$: the elements of both $A$ and $B$.
$A' \cap B$: the elements of $B$ only.
$A' \cap B'$: the elements in neither $A$ nor $B$.

The elements that belong to exactly one of the two sets fill the regions $A \cap B'$ and $A' \cap B$; the union $A \cup B$ is made of the three regions inside the circles. When only the number of elements of each region matters, this number is written in round brackets inside the region.

Proposition — Number of Elements of a Union

For two sets $A$ and $B$ of a finite universal set $U$, $$\textcolor{colorprop}{\Card{A \cup B} = \Card{A} + \Card{B} - \Card{A \cap B}}.$$ Adding $\Card{A}$ and $\Card{B}$ counts the elements of $A \cap B$ twice: subtracting $\Card{A \cap B}$ counts them once. In particular:

if $A$ and $B$ are disjoint ($A \cap B = \emptyset$), then $\Card{A \cup B} = \Card{A} + \Card{B}$;
for the complement of $A$ in $U$, $\Card{A'} = \Card{U} - \Card{A}$.

Proof

Write $a$, $b$, $c$ and $d$ for the numbers of elements of the four regions:

The set $A$ is made of the regions $A \cap B'$ and $A \cap B$, so $\Card{A} = a + b$; likewise $\Card{B} = b + c$ and $\Card{A \cap B} = b$. The union $A \cup B$ is made of the three regions inside the circles, so $\Card{A \cup B} = a + b + c$. Then $$\begin{aligned}[t] \Card{A} + \Card{B} - \Card{A \cap B} &= (a + b) + (b + c) - b && \text{(read on the regions)} \\ &= a + b + c && \text{(collect the terms)} \\ &= \Card{A \cup B}. \end{aligned}$$ If $A \cap B = \emptyset$, then $b = 0$ and the formula becomes $\Card{A \cup B} = \Card{A} + \Card{B}$. Finally, $A$ and $A'$ have no element in common and together make up $U$, so $\Card{A} + \Card{A'} = \Card{U}$.

Example

Given $\Card{U} = 40$, $\Card{A} = 21$, $\Card{B} = 16$ and $\Card{A \cap B} = 9$, find $\Card{A \cup B}$, $\Card{A \cap B'}$ and $\Card{(A \cup B)'}$.

Answer

$$\begin{aligned}[t] \Card{A \cup B} &= \Card{A} + \Card{B} - \Card{A \cap B} && \text{(number of elements of a union)} \\ &= 21 + 16 - 9 \\ &= 28. \end{aligned}$$ The set $A$ is made of the regions $A \cap B'$ and $A \cap B$, so $\Card{A \cap B'} = \Card{A} - \Card{A \cap B} = 21 - 9 = 12$. $$\begin{aligned}[t] \Card{(A \cup B)'} &= \Card{U} - \Card{A \cup B} && \text{(complement in } U\text{)} \\ &= 40 - 28 \\ &= 12. \end{aligned}$$ The Venn diagram gathers the four numbers ($B$ only: $16 - 9 = 7$):

Check: $12 + 9 + 7 + 12 = 40 = \Card{U}$.

Method — Filling a Venn Diagram from Counts

Start with the intersection $A \cap B$, the region shared by the two sets.
Subtract to obtain the regions “$A$ only” and “$B$ only”: $\Card{A \cap B'} = \Card{A} - \Card{A \cap B}$, and likewise for $B$.
The region outside both circles holds what is left: $\Card{U}$ minus the three numbers inside the circles.
If the number of a region is unknown, call it $x$ and write the number of every other region in terms of $x$. Once every region is known in terms of $x$, a total given by the statement, usually $\Card{U}$, gives an equation in $x$: the four regions add up to it. This requires every other region to be known, the outside one included; if the statement gives only $\Card{U}$, $\Card{A}$ and $\Card{B}$, with the outside unknown, the regions add up to $\Card{U}$ whatever $x$ is, and one more piece of information is needed.
Check that every region holds a whole number of elements, zero or more (or, when the regions hold percentages, a percentage between $0$ and $100$), and that the four regions add up to $\Card{U}$ (to $100\pourcent$ when the regions hold percentages).

Example

A hiking club has $36$ members. $25$ of them hike in summer, $19$ hike in winter, and every member hikes in at least one of the two seasons. How many members hike in both seasons?

Answer

Let $S$ and $W$ be the sets of the members who hike in summer and in winter, and let $x = \Card{S \cap W}$. Then $\Card{S \cap W'} = 25 - x$ and $\Card{S' \cap W} = 19 - x$. The region outside both circles is empty, since every member hikes in at least one season.

The four regions make up the club: $$\begin{aligned}[t] (25 - x) + x + (19 - x) + 0 &= 36 && \text{(the regions make up } U\text{)} \\ 44 - x &= 36 && \text{(collect the terms)} \\ x &= 8 && \text{(subtract } 36 \text{, add } x\text{)} \end{aligned}$$ Check: the regions hold $25 - 8 = 17$, $8$ and $19 - 8 = 11$ members, and $17 + 8 + 11 = 36$. So $8$ members hike in both seasons.

Skills to practice

Naming a Region
Using the Formula for a Union
Finding an Unknown Number
Solving Problems with a Venn Diagram

III.3 Venn Diagrams with Three Sets

Recappreviously studied — click to reread

A survey often asks several questions at once: which of three sports each pupil plays, which of three languages each one studies. The answers describe three sets $A$, $B$ and $C$ of the same universal set. Their Venn diagram has three circles that overlap in pairs, and the counting method of two sets still works, one region after the other.

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Definition — Three-Set Venn Diagram

Three sets $A$, $B$ and $C$, drawn as three circles that overlap in pairs and share a common central part inside $U$, divide the diagram into eight regions, numbered here from $1$ to $8$. Every element of $U$ lies in exactly one region.

Region	Set	Its elements are
$1$	$A \cap B' \cap C'$	in $A$ only
$2$	$A' \cap B \cap C'$	in $B$ only
$3$	$A' \cap B' \cap C$	in $C$ only
$4$	$A \cap B \cap C'$	in $A$ and $B$, not in $C$
$5$	$A \cap B' \cap C$	in $A$ and $C$, not in $B$
$6$	$A' \cap B \cap C$	in $B$ and $C$, not in $A$
$7$	$A \cap B \cap C$	in all three sets
$8$	$A' \cap B' \cap C'$	in none of the sets

The regions $1$, $2$, $3$ hold the elements of exactly one set; the regions $4$, $5$, $6$ those of exactly two sets. An intersection of two sets is made of two regions: $A \cap B$ is made of the regions $4$ and $7$.

Example

Let $U = \{1, 2, 3, \dots, 12\}$, let $A$ be the set of the even numbers of $U$, $B$ the set of the multiples of $3$ in $U$, and $C$ the set of the factors of $12$. List $A$, $B$ and $C$, then place the twelve elements of $U$ on a Venn diagram.

Answer

$A = \{2, 4, 6, 8, 10, 12\}$, $B = \{3, 6, 9, 12\}$ and $C = \{1, 2, 3, 4, 6, 12\}$. Each element is placed by asking three questions: is it in $A$? in $B$? in $C$?

$6$ and $12$ belong to the three sets: centre region.
Region “$A$ and $B$, not $C$” stays empty: the even multiples of $3$ in $U$, $6$ and $12$, are both factors of $12$. An empty region is still drawn.
$2$ and $4$ are in $A$ and $C$ only; $3$ is in $B$ and $C$ only.
$8$ and $10$ are in $A$ only, $9$ in $B$ only, $1$ in $C$ only; $5$, $7$ and $11$ are in none of the sets.

Check: $2 + 1 + 1 + 0 + 2 + 1 + 2 + 3 = 12$ elements.

Method — Filling a Three-Set Venn Diagram

Start from the centre, the region $A \cap B \cap C$. If its number is unknown, call it $x$.
Then the three regions “exactly two”: from each intersection of two sets, subtract the centre; for instance $\Card{A \cap B \cap C'} = \Card{A \cap B} - \Card{A \cap B \cap C}$.
Then the three regions “one set only”: from each set, subtract the three regions already filled inside its circle.
Finally the outside: $\Card{U}$ minus the seven numbers inside the circles. When $x$ is unknown and the outside is given by the statement, this last step gives the equation: the eight regions add up to $\Card{U}$.
Check that every region holds a whole number of elements, zero or more (or a percentage between $0$ and $100$ when the regions hold percentages).

When the statement gives a region directly, such as “$A$ and $B$ only”, it is written in its region at once.

Example

In a group of $50$ pupils, $24$ study French, $20$ study Spanish and $18$ study German. $8$ study French and Spanish, $7$ study French and German, $6$ study Spanish and German (each of these three counts includes the pupils who study all three languages), and $3$ study all three languages. How many pupils study none of these languages? Exactly one of them? Exactly two of them?

Answer

Let $F$, $S$ and $G$ be the sets of the pupils who study French, Spanish and German. The centre holds $\Card{F \cap S \cap G} = 3$. The regions “exactly two”: $$\Card{F \cap S \cap G'} = 8 - 3 = 5, \quad \Card{F \cap S' \cap G} = 7 - 3 = 4, \quad \Card{F' \cap S \cap G} = 6 - 3 = 3.$$ The regions “one language only”: $$\begin{aligned}[t] \Card{F \cap S' \cap G'} &= 24 - 3 - 5 - 4 = 12 && \text{(subtract the regions of } F \text{ already filled)} \\ \Card{F' \cap S \cap G'} &= 20 - 3 - 5 - 3 = 9 \\ \Card{F' \cap S' \cap G} &= 18 - 3 - 4 - 3 = 8 \end{aligned}$$ Inside the circles: $12 + 9 + 8 + 5 + 4 + 3 + 3 = 44$ pupils, so $50 - 44 = 6$ pupils are outside.

So $6$ pupils study none of these languages, $12 + 9 + 8 = 29$ study exactly one, and $5 + 4 + 3 = 12$ study exactly two. Check: $6 + 29 + 12 + 3 = 50$.

Example

The members of a sports centre were asked which facilities they had used during the last month. $45\pourcent$ had used the pool, $40\pourcent$ the gym and $30\pourcent$ the courts; $15\pourcent$ had used the pool and the gym, $10\pourcent$ the pool and the courts, $12\pourcent$ the gym and the courts (each of these three percentages includes the members who used all three facilities), and $20\pourcent$ had used none of them. What percentage of the members had used all three facilities?

Answer

Let $P$, $G$ and $C$ be the sets of the members who used the pool, the gym and the courts, and write the percentage of the members in each region. All the members together make $100\pourcent$. Let $x$ be the percentage in the centre $P \cap G \cap C$. The regions “exactly two” hold $15 - x$, $10 - x$ and $12 - x$; then, for the pool only, $$\begin{aligned}[t] 45 - x - (15 - x) - (10 - x) &= 45 - x - 15 + x - 10 + x && \text{(remove the brackets)} \\ &= 20 + x, \end{aligned}$$ and likewise $40 - x - (15 - x) - (12 - x) = 13 + x$ for the gym only and $30 - x - (10 - x) - (12 - x) = 8 + x$ for the courts only.

The eight regions make up all the members: $$\begin{aligned}[t] (20 + x) + (13 + x) + (8 + x) + (15 - x) + (10 - x) + (12 - x) + x + 20 &= 100 && \text{(the regions make up } U\text{)} \\ 98 + x &= 100 && \text{(collect the terms)} \\ x &= 2 && \text{(subtract } 98\text{)} \end{aligned}$$ Check: the regions hold $22$, $15$, $10$, $13$, $8$, $10$, $2$ and $20$, which add up to $100$. So $2\pourcent$ of the members used all three facilities.

Skills to practice

Naming a Region of a Three-Set Diagram
Counting in a Three-Set Venn Diagram
Solving Problems with Three Sets

IV Going further

Skills to practice

Solving Problems with Sets and Venn Diagrams
Quiz

Interval Notation	Set-builder notation	Number line representation
\(\CloseBracketLeft a, b \CloseBracketRight\)	\(\{x \in \R \mid a \leq x \leq b\}\)
\(\CloseBracketLeft a, b \OpenBracketRight\)	\(\{x \in \R \mid a \leq x < b\}\)
\(\OpenBracketLeft a, b \CloseBracketRight\)	\(\{x \in \R \mid a < x \leq b\}\)
\(\OpenBracketLeft a, b \OpenBracketRight\)	\(\{x \in \R \mid a < x < b\}\)
\(\CloseBracketLeft a, +\infty\OpenBracketRight \)	\(\{x \in \R \mid a \leq x\}\)
\(\OpenBracketLeft a, +\infty \OpenBracketRight \)	\(\{x \in \R \mid a < x\}\)
\(\OpenBracketLeft -\infty, a\CloseBracketRight\)	\(\{x \in \R \mid x \leq a\}\)
\(\OpenBracketLeft -\infty, a\OpenBracketRight\)	\(\{x \in \R \mid x < a\}\)

Notation	Meaning	Venn Diagram
\(A\)	Set \(A\)
\(A'\)	Complement of \(A\) (everything in \(U\) not in \(A\))
\(A \subseteq B\)	\(A\) is a subset of \(B\)
\(A \cup B\)	Union of \(A\) and \(B\) (all elements in \(A\) or \(B\))
\(A \cap B\)	Intersection of \(A\) and \(B\) (elements in both)
\(A \cap B = \emptyset\)	\(A\) and \(B\) are disjoint (no common elements)

Region	Set	Its elements are
\(1\)	\(A \cap B' \cap C'\)	in \(A\) only
\(2\)	\(A' \cap B \cap C'\)	in \(B\) only
\(3\)	\(A' \cap B' \cap C\)	in \(C\) only
\(4\)	\(A \cap B \cap C'\)	in \(A\) and \(B\), not in \(C\)
\(5\)	\(A \cap B' \cap C\)	in \(A\) and \(C\), not in \(B\)
\(6\)	\(A' \cap B \cap C\)	in \(B\) and \(C\), not in \(A\)
\(7\)	\(A \cap B \cap C\)	in all three sets
\(8\)	\(A' \cap B' \cap C'\)	in none of the sets