IGCSE, Edexcel, IB and AP Science Courses | Dr Dabour Hub Academy

Ordered pairs and the Cartesian product

An ordered pair (a, b) is a pair of elements in which the order matters.
The first element is a and the second element is b.
So (2, 5) and (5, 2) are different points, and (a, b) = (c, d) only when a = c and b = d.

A pegboard shows the Cartesian product of sets A and B with 3 x 2 = 6 ordered pairs, as relations and functions.

The Cartesian product of two sets X and Y is the set of all ordered pairs whose first element comes from X and whose second element comes from Y:
X × Y = {(x, y) x in X and y in Y}
If X has 3 elements and Y has 4 elements, then X × Y has 3 × 4 = 12 ordered pairs.
In general, n(X × Y) = n(X) × n(Y).
The product is not commutative.
X × Y and Y × X contain pairs with the coordinates swapped, so they are equal only when X = Y or one of the sets is empty.

Three ways to show a Cartesian product

Listwrite all the pairs inside braces.

Tableput the elements of X down the side and the elements of Y across the top, then fill each cell with the pair.

Griddraw the elements of X on the horizontal axis and the elements of Y on the vertical axis, then mark a point at every pair.

Lesson 1 – Cartesian Product, Relations, Functions and Quadratic Graphing · 1 / 7

Relations, domain, codomain and range

A relation R from a set X to a set Y is any subset of X × Y.
It connects some elements of X with some elements of Y.
The three important sets for a relation are:

Domain the set of all first elements of the pairs in R. It is a subset of X.
Codomain the set Y that the relation maps into.
Range the set of all second elements of the pairs in R. It is a subset of Y.
A relation can be shown as a set of pairs, as an arrow diagram, or as points on a grid.
In an arrow diagram, each arrow goes from an element of X to an element of Y.
Some elements of Y may receive no arrow, and an element of X may send more than one arrow.

Lesson 1 – Cartesian Product, Relations, Functions and Quadratic Graphing · 2 / 7

Functions and the function test

A relation from X to Y is a function (also called a mapping) when every element of X is the first element of exactly one ordered pair.
In the arrow diagram, this means that each element of X has exactly one outgoing arrow.
Important the test checks only the set X. Elements of Y may receive several arrows, or none, and the relation is still a function. Only an element of X with two outputs, or with no output at all, breaks the rule.
A function is usually written as y = f(x). The value f(x) is the image of x, and the set of all images is the range.

Linear and quadratic functions

A linear function f(x) = mx + c graphs as a straight line.
A quadratic function f(x) = ax² + bx + c, where a is not zero, graphs as a parabola.
The main facts about a parabola are:

Comparison aspect	Main idea	Keywords
Vertex	the point (−b ÷ 2a f(−b ÷ 2a))	point
Axis of symmetry	the vertical line x = −b ÷ 2a through the vertex	verticallinethroughvertex
Direction	if a > 0 the parabola opens upward and the vertex is the minimum point if a < 0 it opens downward and the vertex is the maximum point	parabolaopensupwardvertex

Lesson 1 – Cartesian Product, Relations, Functions and Quadratic Graphing · 3 / 7

Graphing quadratic functions with a table

To draw a quadratic graph, find the vertex first and then choose x-values on both sides of it.
Include the endpoints when the question gives an interval.
A small table is the most reliable method:

x	−1	0	1	2	3
f(x) = x² − 2x	3	0	−1	0	3

The values are symmetric around x = 1, which is the axis of symmetry.
The lowest value, −1, is the minimum, and it occurs at the vertex (1, −1).
Two checks help you avoid mistakes.
First, the values should be symmetric around the axis.
Second, the vertex value must be the smallest value in the table when a is positive, or the largest when a is negative.
The x-intercepts, where f(x) = 0, appear on either side of the axis when the vertex is below the x-axis.

Lesson 1 – Cartesian Product, Relations, Functions and Quadratic Graphing · 4 / 7

Worked examples

Comparison aspect	Main idea	Keywords
Example 1 (Cartesian product).	X = {a b} and Y = {1 3}. Find X × Y and n(X × Y)	Find
Solution	X × Y = {(a 3)}. Since n(X) = 2 and n(Y) = 3 n(X × Y) = 2 × 3 = 6 which agrees with the list	Sinceagreeslist
Example 2 (equal ordered pairs).	If (x + 2 3y − 1) = (5 11) find x and y	find
Solution	Equal ordered pairs have equal first elements and equal second elements. So x + 2 = 5 gives x = 3 and 3y − 1 = 11 gives y = 4	Equalorderedpairsfirst
Example 3 (a function).	X = {1 4} and Y = {1 6}. The relation R is defined by y = x + 2. Write R decide whether it is a function and state the domain and range	relationdefinedWritedecide

Comparison aspect	Main idea	Keywords
Solution	R = {(1 6)}. Each element of X appears exactly once as a first element so R is a function. The domain is {1 and the range is {3	elementappearsexactlyonce
Example 4 (not a function).	S = {(−1 2)} is a relation from X = {−1 1} to Y = {0 2}. Decide whether S is a function	relationDecidewhether
Solution	The element 1 of X is the first element of two pairs 0) and (1 2). One input has two outputs so S is not a function. It does not matter that the output 0 comes from two different inputs because the test looks only at the inputs	elementfirstpairsinput
Example 5 (vertex and axis).	Find the vertex and axis of symmetry of f(x) = x² − 6x + 8 then sketch the table of values for 0 ≤ x ≤ 6	Findsymmetrythensketch

Comparison aspect	Main idea	Keywords
Solution	a = 1 and b = −6 so the axis is x = −(−6) ÷ 2 = 3. Then f(3) = 9 − 18 + 8 = −1. The vertex is (3 −1). The table is: x = 0 6 gives f = 8 8. The parabola opens upward because a = 1 > 0 so the minimum value is −1	axisThenvertextable
Example 6 (maximum value).	Find the maximum value of f(x) = −2x² + 8x − 3 and the value of x where it occurs	Findoccurs
Solution	a = −2 is negative so the parabola opens downward and the vertex is the maximum point. The axis is x = −8 ÷ (2 × (−2)) = 2. Then f(2) = −8 + 16 − 3 = 5. The maximum value is 5 at x = 2	negativeparabolaopensdownward
Example 7 (range on an interval).	For f(x) = 2x² − 4x + 5 find the range of f when −1 ≤ x ≤ 3	find

Solution→The vertex is at x = 4 ÷ 4 = 1, which lies inside the interval. f(1) = 2 − 4 + 5 = 3, and this is the minimum. At the endpoints, f(−1) = 2 + 4 + 5 = 11 and f(3) = 18 − 12 + 5 = 11. So the range is all values from 3 to 11, written [3, 11].

Lesson 1 – Cartesian Product, Relations, Functions and Quadratic Graphing · 5 / 7

Common mistakes

Reversing the coordinates. The point (2, 5) is not the same as (5, 2), so write the first element from the first set.
Checking the codomain instead of the domain. The function test looks at each element of X, not at how often each element of Y is used.
Writing the axis as x = b ÷ 2a. The correct formula is x = −b ÷ 2a, so the sign of b changes.
Choosing the wrong extreme. When a is positive the vertex is a minimum, and when a is negative it is a maximum.
Leaving out the endpoints of an interval when building a table or finding a range.

Lesson 1 – Cartesian Product, Relations, Functions and Quadratic Graphing · 6 / 7

Lesson summary

An ordered pair (a, b) has order; (a, b) = (c, d) means a = c and b = d.
X × Y is the set of all pairs with the first element from X and the second from Y, and n(X × Y) = n(X) × n(Y).
A relation is a subset of X × Y. Its domain is the set of first elements and its range is the set of second elements.
A relation is a function when each element of X appears exactly once as a first element.
The graph of f(x) = ax² + bx + c is a parabola with vertex (−b ÷ 2a, f(−b ÷ 2a)) and axis of symmetry x = −b ÷ 2a.
If a is positive, the vertex is a minimum. If a is negative, the vertex is a maximum.

Lesson 1 – Cartesian Product, Relations, Functions and Quadratic Graphing · 7 / 7

Curriculum

Mathematics — Third Preparatory

Unit 1

Unit 2

Unit 3

Unit 4

Lesson 1 – Cartesian Product, Relations, Functions and Quadratic Graphing