What must match before two ordered pairs count as equal?

Both entries in their positions. Equal pairs match first with first and second with second.

Which order of the factors A and B contains the pair (0,2)?

A× B The reversed product cannot start with 0 because B does not contain it.

Which product of the named sets contains only (0,2)?

A× C Its first entry comes from A and its second from C.

Which codomain member is missing from this range?

0. No selected pair has 0 as its second entry.

Why does the shared output 1 still allow this relation to be a function?

Each input has one output. The condition counts outputs per input, not inputs per output.

What output does this reciprocal rule give for negative one?

Negative one. Dividing a positive number by a negative number gives a negative result.

What does modulus return for negative two?

2. Negative two lies two units from zero.

What remains when x plus one is subtracted from x?

−1. The x terms cancel, leaving the subtraction of one.

Teach Relations and Functions in eight steps.

For each idea: show it, ask before you tell, then say one line.

Gather paper and a pencil.

Common misconceptions & worked visual for this chapter · 15 practice questions with answers

Why it matters

Clothing choices combine to make outfits.
Library catalogues connect books with their details so readers can find them.
Subtracting costs from income shows how much money remains.

What to teach

How to teach it

Pairs and products

Four points show every pairing of first coordinates with second coordinates.

1. Ordered pairs and Cartesian product of sets

Ordered pairs keep positions; a Cartesian product includes every combination.

Show: Look at the diagram. Read each point's horizontal coordinate first, then its vertical coordinate. Collect the different entries in each position.
Ask: If we rebuilt this product, which entries would supply A and which would supply B?
Say: Each first coordinate pairs with every second coordinate. For this nonempty product, first coordinates recover A, and second coordinates recover B.
Used in: Meal choices combine mains with accompaniments, helping kitchens list available combinations.

If your child says…“Changing the order makes no difference.”

Ask: Would swapping 1 and 3 keep their point in the same position?
What’s true: Switching different entries changes an ordered pair.

$A\times B=\{(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)\}$

$n(A\times B)=2\times3=6$

$n(A\times B\times C)=6\times2$

2. Number of elements in a Cartesian product and ordered triples

Multiply choices across positions.

Show: Say this first, then write the formula: A contains 0 and 1. B contains 2, 3 and 4. C contains 5 and 6.
Ask: How many triples can these six pairs make with C?
Say: There are twelve triples because each pair has two choices from C.
Used in: Map coordinates help locate destinations.

If your child says…“The counts match, so the products match.”

Ask: Can the reversed product start a pair with 0?
What’s true: Reversed products coincide only when their factors match or either factor is empty.

$A\times(B\cup C)=\{0\}\times\{1,2\}$

$A\times(B\cup C)=\{(0,1),(0,2)\}$

$A\times(B\cup C)=\{(0,1)\}\cup\{(0,2)\}$

3. Cartesian product and set operations

The outside factor pairs with both inside sets.

Show: First say: A contains 0, B contains 1, and C contains 2. Union collects members from either set. Then write these lines.
Ask: Which products would give these two pairs separately?
Say: A times B gives (0,1), and A times C gives (0,2). Their union gives the same pairs as pairing A with B union C.
Used in: Combining activity lists preserves registrations without inventing new student–activity pairs.

If your child says…“Only the first inside set needs A.”

Ask: Where would the first entry of the second pair come from without A?
What’s true: Both resulting products need the outside factor.

Relations and functions

One arrow connects domain element zero to range element one within the larger codomain.

4. Relations; domain, codomain and range

Selected pairs determine participating inputs and outputs.

Show: Look at the diagram for 'first entry smaller than second'. Follow the arrow, then inspect all of B.
Ask: Which entries take part under this rule?
Say: The domain contains 0, the range contains 1, and the codomain contains both.
Used in: Library borrowing records help track returns.

If your child says…“The range includes everything in B.”

Ask: Which arrow reaches 0?
What’s true: The range includes reached targets, while the codomain includes every allowed target.

Three inputs each have one arrow, with two inputs sharing an output.

5. Functions

Every allowed input receives exactly one output.

Show: Look at the diagram. Count the arrows leaving each member of A. Follow the arrows from a and b to their shared output.
Ask: Does choosing any input here determine exactly one answer?
Say: Yes, every input has exactly one outgoing arrow, so this relation is a function. Different inputs may share an output.
Used in: A weighing scale assigns a weight reading to each parcel, helping calculate postage.

If your child says…“Different inputs need different outputs.”

Ask: Does a gain another output just because b reaches the same place?
What’s true: A function permits shared outputs but requires exactly one output for every input.

Working with function rules

$f(x)=\frac{1}{x}$

$f(-1)=\frac{1}{-1}=-1$

$\operatorname{Dom}(f)=\mathbb{R}\setminus\{0\}$

6. Evaluation, domain and range of a real function

Check allowed inputs and attainable outputs.

Show: Say this first, then write the formula: f divides one by input x. Division by zero is undefined, so zero cannot be an input.
Ask: Which real numbers can come out of this rule?
Say: Every nonzero real number can come out, so the range also excludes zero. One divided by a nonzero number never gives zero. Any nonzero output is reached by using its reciprocal as the input.
Used in: Fare rules help travellers budget.

If your child says…“I can ignore the input’s negative sign.”

Ask: What number multiplied by negative one gives positive one?
What’s true: Substitution keeps the input’s sign.

A V-shaped modulus graph shows opposite inputs reaching the same positive height.

7. Some functions and their graphs

Rules determine graph shapes.

Show: Look at the diagram: x marks inputs, and y marks outputs. Find the marked inputs, negative two and two. Follow their points to the same height.
Ask: From this graph, what output would negative three give?
Say: Negative three gives three because it lies three units from zero.
Used in: Size deviations help factories check accuracy.

If your child says…“A negative input stays negative inside the bars.”

Ask: Can a distance below zero describe how far apart two points are?
What’s true: Modulus measures distance from zero, so its output cannot be negative.

$(f-g)(x)=f(x)-g(x)$

$(f-g)(x)=x-(x+1)$

$(f-g)(x)=x-x-1$

8. Algebra of real functions

Combine outputs at the same input.

Show: Say this first, then write the formula: f returns x; g returns x plus one. Subtract the whole output.
Ask: What remains after cancellation?
Say: Negative one remains.
Used in: Subtracting costs from income shows profit.

If your child says…“The minus changes only the first term.”

Ask: Have you subtracted the whole second output if you add its one?
What’s true: Subtracting a bracket subtracts every term inside.

A real problem

Mathematical relations made stored information easier to access.

Edgar Codd introduced the relational database model in 1970.

Earlier, retrieving information commonly required specialists to write programs.

His model organised records using mathematical relations involving several sets.

Relational databases made stored information easier for ordinary users to access.

ACM SIGMOD IBM Wikipedia

A library computer displaying a searchable book catalogue.

Concept Lab · Common misconception

The tempting mistake in Relations and Functions

Open standalone misconception page →

Can a function send one input to two outputs?

Wrong

The relation {(1, 2), (1, 3)} is a function.

Why it feels right. Both pairs follow the same input-output format.

Right

A function assigns exactly one output to each input in its domain.

Worked example

Input 1 has two outgoing arrows to distinct outputs, so this relation fails the function rule.

On an arrow diagram, input 1 has arrows to both 2 and 3.
That violates the one-output rule.

Ask your child eight quick questions.

One per step. The answer and the reason are given.

What must match before two ordered pairs count as equal?

Both entries in their positions. Equal pairs match first with first and second with second.
Which order of the factors A and B contains the pair (0,2)?

$A\times B$ The reversed product cannot start with 0 because B does not contain it.
Which product of the named sets contains only (0,2)?

$A\times C$ Its first entry comes from A and its second from C.
Which codomain member is missing from this range?

0. No selected pair has 0 as its second entry.
Why does the shared output 1 still allow this relation to be a function?

Each input has one output. The condition counts outputs per input, not inputs per output.
What output does this reciprocal rule give for negative one?

Negative one. Dividing a positive number by a negative number gives a negative result.
What does modulus return for negative two?

2. Negative two lies two units from zero.
What remains when x plus one is subtracted from x?

−1. The x terms cancel, leaving the subtraction of one.

15 more questions with answers, 2 with pictures

Teach Relations and Functions in eight steps.

Why it matters