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A set is a collection of distinct characters or elements enclosed within curly brackets and separated by commas. The ordered pair of a coordinate system contains elements that are orderly within the braces. It is the x and y coordinates written within the brackets as (x,y). The concept of the Cartesian product was first introduced by Rene Descartes in 1637. It portrays the product of the ordered elements of two sets. The Cartesian product is not restricted to only two ordered sets. According to the number of sets involved in the Cartesian product, the dimensions of the plane differ.
Ordered Pairs
Interchanging the order of the elements changes the whole meaning of the set.
The x-coordinate should be in the place of x(abscissa) only.
So is the rule for the y-coordinate, too. The ‘y’-coordinate should be in the place of the y-axis(ordinate) value.
Ordered pairs don’t mean that the pairs in the sets are equal to each other.
Ordered pair example
On solving an algebraic equation, the obtained values of
x are 1 and 3
y are 2 and 4
When they are written as a set, they are ordered as
A = (1,2)
B = (3,4)
A Cartesian plane
The Cartesian plane is, generally, a two-dimensional plane consisting of the x and y axes.
The coordinates of the Cartesian coordinate plane are a set of ordered pairs of a coordinate system.
The ordered sets in the Cartesian plane are plotted in the x and y axes in an orderly manner.
The order of the Cartesian coordinates is never altered.
A one-dimensional Cartesian plane
A one-dimensional Cartesian plane shows a straight line drawn like a number line.
It has real numbers on the line, giving the distance of a point from the O.
A two-dimensional Cartesian plane
The coordinates are obtained from the Cartesian product of two ordered sets in the Cartesian plane.
A two-dimensional Cartesian plane is a graph drawn with an ordered pair of a coordinate system.
The ordered pair has two values of x-coordinate and y-coordinate, forming a 2D graph.
The graph has two perpendicular lines of x and y axes that intersect each other at 0.
The x-coordinate is called abscissa, and the ‘y’-coordinate is the ordinate.
A three-dimensional Cartesian plane
The coordinates are the Cartesian product of three ordered sets in the Cartesian plane.
It gives a three-dimensional graph in the Cartesian coordinate system.
The ordered triplets have three values for the x, y, and z coordinates, forming a 3D graph.
The graph has three perpendicular lines intersecting each other at right angles in the point of origin O.
Cartesian product
The Cartesian product of ordered sets is denoted as a multiplication function of the two given sets. For example, considering the ordered sets X and Y, the Cartesian product is denoted as X x Y.
Cartesian product example
Considering the examples given in the above topic ‘Ordered pair example’, the ordered pair of sets are
Set A = {1,2}
Set B = {3,4}
Their Cartesian product is
A x B = {(1,3)(2,4)(2,3)(1,4)}
Where,(1,2) ∈ A
(3,4) ∈ B
Their orders cannot be changed because changing the order of the ordered sets makes the sum faulty.
Solved examples
Here are some solved examples for a better understanding of the ordered sets in a Cartesian plane.
Consider a set where A = {a,b} and B = {y,z}. Write the Cartesian product of the ordered pair.
A = {a,b}
B = {y,z}
The Cartesian product is denoted as A x B.
A x B = {(a,y)(b,z)(a,z)(b,y)}
Consider a set Y = {white,black} and Z = {chalk,pencil}. Write the Cartesian product of the given ordered sets.
Y = {white,black}
Z = {chalk,pencil}
The Cartesian product is denoted as X x Y.
X x Y = {(white,chalk)(white,pencil)(black,chalk)(black,pencil)}
Consider sets A and B. The Cartesian product of set A x B has 50 elements, and set A has ten elements. How many elements are present in set B?
The product of the number of elements in the given two sets gives the total number of elements in the set Cartesian product.
Number of elements in A x B = 50
Number of elements in A = 10
B = (A x B) / A
B = 50 / 10
B = 5
There are 5 elements in the ordered set B.
If the Cartesian product set of A x B is {(1,5),(9,5)}. Find the ordered sets A and B.
The Cartesian product set has the elements of the individual sets clubbed in an ordered manner.
A x B = {(1,5),(9,5)}
A = {1,9}
B = {5}
Conclusion
In the ordered pairs of a coordinate system, their order cannot be interchanged. The abscissa lies in the x-axis, and the ordinate lies in the y-axis. Their set product is given by the Cartesian product. The pairs of the set lie as the ordered sets in the Cartesian plane. Based on the sets considered for the Cartesian product, the Cartesian plane differs as one or two or three or multidimensional. When two ordered pairs are considered, the Cartesian plane is two-dimensional. When three ordered triplets are considered, the Cartesian plane is three-dimensional. You have learnt about ordered sets and ordered sets in Cartesian planes in this article.
