Lesson 1 ~35 min Functions · Path +80 XP

Relations and Ordered Pairs

Every graph you have drawn, every table you have filled in and every rule you have written as $y = \ldots$ was describing the same kind of object. This lesson gives that object a name and a precise notation, which is what the rest of the focus area is built on.

Today's hook: A set of ordered pairs, a table of values, a mapping diagram and a scatter of points on a grid can all describe exactly the same thing. Learning to move between the four is most of what this lesson asks, and it is the reason function notation later feels like shorthand rather than a new idea.

0/5QUESTS

Think First

warm-up

Write down the pairs $(1,2)$ and $(2,1)$. Are they the same pair? Now think of a real situation the pairing could describe, such as "student to locker number", and say what changes if you swap the two entries. That is why the word ordered is in ordered pair.

Record your answer in your workbook.

The Big Idea

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A relation between two sets is any association pairing elements of the first set with elements of the second. Written out, it is a set of ordered pairs, and the first entry always comes from the first set.

$$R = \{(1,2),\ (2,4),\ (3,6)\}$$

The same relation can be written four ways: as a set of ordered pairs, as a table of values, as a mapping diagram, or as points on a number plane. None is more correct than the others, and being able to move between them is the whole skill of this lesson.

$\{(x,y)\}$

Order matters

$(1,2)$ and $(2,1)$ are different. The first entry is always the input side.

Any association counts

A relation needs no formula and no pattern. A list of pairs is already a relation.

Use set brackets

A relation is a set, so it is written inside curly brackets with the pairs separated by commas.

What You'll Master

objectives

Know

That a relation over two sets associates elements of the first with elements of the second
That a relation can be written as a set of ordered pairs
That the order within a pair is part of its meaning

Understand

Why a relation needs no rule or formula to exist
Why four different representations can describe one relation

Can Do

Write a relation as a set of ordered pairs
Convert between ordered pairs, a table, a mapping diagram and a graph
Read a relation off a described situation

Words You Need

vocabulary

SetA collection of distinct objects, written inside curly brackets.

ElementOne member of a set.

RelationAn association between the elements of one set and those of another.

Ordered pairTwo values written as $(a,b)$, where the order carries meaning.

Mapping diagramTwo sets drawn as regions, with arrows showing the pairings.

Number planeThe grid on which an ordered pair $(x,y)$ is plotted as a point.

What a Relation Is

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Take two sets. A relation between them is any rule, list or association that pairs elements of the first set with elements of the second.

That definition is deliberately generous. A relation does not need a formula, a pattern or any regularity at all. This is a perfectly good relation:

$$R = \{(1, 7),\ (1, -3),\ (4, 7),\ (9, 0)\}$$

No rule produces those pairs, and none is needed. Listing them is enough, because a relation just is its set of pairs.

Relations do not have to be about numbers either. "Student is enrolled in subject" is a relation between the set of students and the set of subjects. In this focus area both sets will always be sets of real numbers, so every pair is a pair of numbers and can be plotted, but the general idea is wider than that.

Why start so broadly

Functions are a special kind of relation, and the special condition only makes sense once you know what it is a condition on. Lesson 2 adds that condition. Everything here is the container it goes in.

Ordered Pairs, and Why Ordered

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Each pairing is recorded as an ordered pair, written $(a, b)$ with round brackets and a comma.

The order is part of the meaning. $(1, 2)$ says "the element $1$ from the first set is paired with the element $2$ from the second". $(2, 1)$ says something different, and in general a relation containing one need not contain the other.

Compare with a set, where order carries no meaning at all: $\{1, 2\}$ and $\{2, 1\}$ are the same set. Curly brackets discard order; round brackets keep it. The relation itself is a set of pairs, so the pairs inside it may be listed in any order, but no individual pair may be rewritten backwards:

$$\{(1,2),\ (3,6)\} = \{(3,6),\ (1,2)\} \quad \text{but} \quad \{(1,2)\} \neq \{(2,1)\}$$

On the number plane the two brackets do different work too. The point $(1,2)$ is one across and two up; the point $(2,1)$ is two across and one up. They are different points, which is the same fact drawn rather than written.

Four Ways to Write One Relation

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Consider the relation pairing each of $1$, $2$ and $3$ with its double.

As a set of ordered pairs.

$$\{(1,2),\ (2,4),\ (3,6)\}$$

As a table of values. One row for the first entries and one for the second, with the columns lining the pairs up.

As a mapping diagram. Two regions with arrows between them, as in the diagram above. This one shows most clearly what is paired with what, and it will be the easiest way to see the function condition in the next lesson.

As a graph. Three points plotted at $(1,2)$, $(2,4)$ and $(3,6)$ on the number plane.

All four describe the same object. Which to use depends on the question: a mapping diagram makes the pairings visible, a graph makes the pattern visible, and a set of pairs is the one that is unambiguous and so is used in definitions.

Relations Given by a Rule

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When there is a rule, the relation can be described without listing anything. The set-builder form reads as a sentence:

$$\{(x,y) : y = 2x\}$$

which says "the set of all pairs $(x,y)$ such that $y = 2x$". The colon is read as "such that". Since $x$ may be any real number, this relation contains infinitely many pairs and its graph is the whole straight line $y = 2x$, not three isolated dots.

Two relations can share a rule and still differ, if the allowed inputs differ:

$$\{(x,y) : y = 2x\} \qquad \text{and} \qquad \{(x,y) : y = 2x, \ x = 1, 2, 3\}$$

The first is a line; the second is three points on that line. The rule alone does not determine the relation, and the set of allowed inputs is the other half of the description. That set has a name, the domain, and Lesson 4 is about it.

An equation such as $x^2 + y^2 = 25$ also defines a relation, whose graph is a circle. Notice that it is not written in the form $y = \ldots$, and that nothing in the definition requires it to be.

Common Pitfalls

+5 XP to read

Writing a pair backwards, so that $(3,6)$ becomes $(6,3)$.

Fix: the first entry always comes from the first set, and on a graph it is the horizontal coordinate. Read the pair left to right and plot across then up.

Assuming a relation must have a formula.

Fix: a list of pairs with no pattern is still a relation. The definition asks for an association, not for a rule.

Using curly brackets for a pair, writing $\{3,6\}$ instead of $(3,6)$.

Fix: curly brackets make a set, which has no order. A pair needs round brackets so that the first and second entries stay distinguishable.

Treating a relation given by a rule as though it had only the few points you happened to plot.

Fix: unless the allowed inputs are restricted, the relation contains every pair satisfying the rule. Join the points and say what the whole graph is.

Watch Me Solve It · 3 examples

Watch Me Solve It · Writing a relation four ways

+15 XP per step

PROBLEM

A relation pairs each of the numbers $-1$, $0$ and $2$ with its square. Write it as a set of ordered pairs, describe its mapping diagram, and state the points on its graph.

1

Apply the rule to each first entry

$(-1)^2 = 1, \qquad 0^2 = 0, \qquad 2^2 = 4$

The first entry is the number, the second is its square.
2

Write the set of ordered pairs

$\{(-1, 1),\ (0, 0),\ (2, 4)\}$

Round brackets for each pair, curly brackets around the whole set.
3

Describe the mapping diagram

The left region holds $-1$, $0$ and $2$; the right region holds $1$, $0$ and $4$. Three arrows run from $-1$ to $1$, from $0$ to $0$, and from $2$ to $4$.
4

State the graph

Three isolated points at $(-1,1)$, $(0,0)$ and $(2,4)$. They are not joined, because only three first entries were allowed.

Answer$\{(-1,1),\ (0,0),\ (2,4)\}$, plotted as three separate points

Watch Me Solve It · Reading a relation from a description

+15 XP per step

PROBLEM

In a class, Ali studies Maths and Music, Bea studies Maths, and Cal studies Music and Art. Write the relation "student studies subject" as a set of ordered pairs, and state how many pairs it contains.

1

Identify the two sets

The first set is $\{\text{Ali}, \text{Bea}, \text{Cal}\}$ and the second is $\{\text{Maths}, \text{Music}, \text{Art}\}$. The relation runs from students to subjects, in that order.
2

Write one pair per stated fact

$(\text{Ali}, \text{Maths}), \ (\text{Ali}, \text{Music}), \ (\text{Bea}, \text{Maths}), \ (\text{Cal}, \text{Music}), \ (\text{Cal}, \text{Art})$

A student studying two subjects contributes two pairs, one for each.
3

Count them

Five pairs. Note that Ali appears as a first entry twice, which the definition of a relation permits without any difficulty.
4

Notice what this will matter for

In Lesson 2 a function will require each first entry to appear exactly once. This relation fails that, so it is a relation but not a function. Recognising it now costs nothing and saves work later.

AnswerFive pairs, with Ali appearing twice as a first entry

Watch Me Solve It · A relation given by a rule

+15 XP per step

PROBLEM

Consider $R = \{(x,y) : y = x + 3\}$. (a) State three pairs belonging to $R$. (b) Is $(2, 6)$ in $R$? (c) Describe the graph of $R$.

1

(a) Choose any first entries and apply the rule

$(0,3), \quad (1,4), \quad (-5,-2)$

Any real number may be used as the first entry, since nothing restricts it.
2

(b) Test the candidate pair against the rule

$x = 2 \ \Rightarrow \ y = 2 + 3 = 5 \neq 6$

The pair $(2,6)$ does not satisfy $y = x + 3$, so it is not in $R$. The pair $(2,5)$ is.
3

(c) Describe the whole set

Every real $x$ gives exactly one pair, so $R$ contains infinitely many pairs. Its graph is the entire straight line $y = x + 3$, with gradient $1$ and $y$-intercept $3$.
4

Say why it is a line and not points

Nothing restricts the first entries, so no gaps appear. Had the description added "where $x$ is a whole number", the graph would have been a row of isolated dots on that line instead.

Answer(a) e.g. $(0,3)$, $(1,4)$, $(-5,-2)$; (b) no, since $(2,5)$ is the pair with first entry $2$; (c) the whole line $y = x+3$

Brain Trainer · 5 problems

Brain Trainer · Pairs and relations

5 problems

Five quick items on relations and ordered pairs. Work each one, then reveal the answer.

1 Are $(4, 9)$ and $(9, 4)$ the same ordered pair?

The order carries meaning in a pair.No
2 Write the relation pairing $1$, $2$ and $3$ each with its cube.

Cube each first entry.$\{(1,1), (2,8), (3,27)\}$
3 Is $(-2, 4)$ in the relation $\{(x,y) : y = x^2\}$?

Test it: $(-2)^2 = 4$.Yes
4 How many ordered pairs are in $\{(0,1), (1,0), (0,1)\}$?

A set contains no duplicates, so the repeat counts once.$2$
5 Does the equation $x^2 + y^2 = 9$ define a relation?

It associates values of $x$ with values of $y$; no rule of the form $y = \ldots$ is required.Yes, a circle

Complete in your workbook.

Multiple Choice · 5 questions

MC1

The definition

+10 XP

A relation between two sets is:

MC2

Order

+10 XP

Which statement is TRUE?

MC3

Membership

+10 XP

Which pair belongs to $\{(x,y) : y = 3x - 1\}$?

MC4

Representations

+10 XP

A mapping diagram, a table of values and a set of ordered pairs:

MC5

Restricted inputs

+10 XP

The relation $\{(x,y) : y = 2x, \ x = 0, 1, 2\}$ has a graph that is:

Short Answer · 3 questions

Four representations

+15 XP

SHORT ANSWER

A relation pairs each of $-2$, $-1$, $0$ and $1$ with the number three more than its double.
(a) Write the relation as a set of ordered pairs.
(b) Describe the mapping diagram in words.
(c) State the four points of the graph, and explain why they should not be joined.
(d) Write the relation in set-builder form, including the restriction on the first entries.

Write your working in your book.

Testing membership

+15 XP

SHORT ANSWER

Let $R = \{(x,y) : y = x^2 - 4\}$.
(a) Find the second entry of the pair in $R$ whose first entry is $3$.
(b) Determine whether $(-3, 5)$ is in $R$, showing your working.
(c) Find all pairs in $R$ whose second entry is $0$.
(d) Explain why there is exactly one pair in $R$ for each first entry, but not exactly one for each second entry.

Write your working in your book.

Relations from a context

+15 XP

SHORT ANSWER

A shop records which of its three staff, P, Q and R, can operate which of its two machines, M1 and M2. P can operate both; Q can operate M2 only; R can operate neither.
(a) Write the relation "staff member can operate machine" as a set of ordered pairs.
(b) State how many pairs it has, and explain what R contributing no pairs means.
(c) Write the reverse relation, "machine can be operated by staff member", as a set of ordered pairs.
(d) Explain why the two relations in (a) and (c) are different, even though they record the same information.

Write your working in your book.

Stretch Challenge · How many relations are there

+25 XP

CHALLENGE

(a) Let $A = \{1,2\}$ and $B = \{5,6\}$. List every ordered pair with first entry from $A$ and second entry from $B$.
(b) A relation from $A$ to $B$ is any subset of that list, including the empty one. How many relations from $A$ to $B$ are there? Justify your count.
(c) Generalise: if $A$ has $m$ elements and $B$ has $n$ elements, how many relations are there from $A$ to $B$? Explain each step of your reasoning.

Quick Review

recap

Relation

Any association between two sets

Ordered pair

$(a,b)$: round brackets keep the order

Four notations

Pairs, table, mapping diagram, graph

Rules are optional

A list with no pattern is still a relation

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