Ordered Pair of Sets

An ordered pair constitutes a pair of coordinates that has a constant place. The ordered pairs have x-coordinate and y-coordinate, which has no interchangeable property. The x-coordinate belongs to the x-axis, and the y-coordinate belongs to the y-axis. Just because they are a set of ordered pairs, for example, (x1,y1) and (x2,y2), does not mean that they can be written by alternating their places before and after the comma within the parentheses.

The ordered pair definition clearly states that the elements are present in an ordered manner. The Cartesian product of the sets is calculated from the ordered pair of sets. This article will discuss if an ordered pair constitutes a set.

Characteristics

• An ordered pair is enclosed within the parentheses.
• It contains two elements within the brackets, separated by a comma. 
• The position of the elements in an ordered set never changes.
• The two elements in the ordered set are the x and y coordinates of the set.

X-coordinate 

• The x-coordinate is named otherwise as abscissa. 
• It measures the distance between the x-axis and the x-coordinate.
• It stands for the horizontal measure in the coordinate plane system.
• It is on the left side of the ordered set as the first element, before the comma.

Y-coordinate 

• The ‘y’-coordinate is called the ordinate of the graph point.
• It shows the measure between the y-axis and the ‘y’-coordinate point.
• It denotes the vertical distance in a coordinate plane system.
• It takes its place on the right side of the ordered set as the second element, present after the comma.

Set of ordered pairs example

• Consider set A and set B.
• The ordered pairs of set A = {a,b}. 
• The ordered pairs of set B = {c,d}.
• The given set of paired elements has a fixed place.
• The Cartesian product of the ordered sets is 

Set of (A x B) = {(a,b),(a,c),(b,d),(b,c)}

Set of ordered pairs in a graph

The set of ordered pairs with varied signs takes place in different quadrants on a graph sheet. Consider the alphabets a and b for the set of ordered pairs. Let us see how they change their sign when placed in each of the four quadrants.

Quadrant 1

• The value of ‘a’ is greater than zero(positive).  
• The value of ‘b’ is also greater than zero(positive).

a > 0; b > 0

Quadrant 2

• The value of ‘a’ is less than zero(negative). 
• The value of ‘b’ is greater than zero(positive).

a < 0; b > 0

Quadrant 3

• The value of ‘a’ is less than zero(negative). 
• The value of ‘b’ is also less than zero(negative).

a < 0; b < 0

Quadrant 4

• The value of ‘a’ is greater than zero(positive). 
• The value of ‘b’ is less than zero(negative).

a > 0; b < 0

Equality Property 

Under the property of equality, the set ordered pairs are equated with the respective coordinates. It helps to solve the unknown variables when it is present in one of the coordinates.

Example

Consider an ordered set pair X = {(x-3),(y+3)} and Y = {(5),(9)}. 

To find the variables x and y in the set X, equate the respective coordinates of the sets.

• (x-3) = (5)

x – 3 = 5

x = 5 + 3

x = 8

• (y+3) = (9)

y+3 = 9

y = 9-3

y = 6

Thus, the property of equality for the set of ordered pairs solves the unknown variables in a set.

Solved Examples

Here are some solved examples on the set of ordered pairs for better understanding and clarity.

• If the given ordered sets are (a,-b)  where a>0 and b>0, then in which quadrant do they lie in a graph?

Given that, the ordered pair of a set is (a,-b).

Since a is positive, it is greater than zero.

Since b is negative, it is less than zero.

Thus, the points lie on the 4th quadrant generally,

x > 0 and y < 0

• Consider the ordered pair of sets A = {a,b} and B = {f,g}. What is the Cartesian product of the ordered set?

Given that, 

A = {a,b} 

B = {f,g}

The Cartesian product of the given ordered set is

A x B = {(a,f),(b,f),(a,g),(b,g)}

• If the number of elements in the Cartesian product of an ordered set is 20, and the number of elements in set B is 4, what is the number of elements present in set A?

Given that, 

A x B = 20

B = 4

Then, 

A = (A x B) / B

A = 20 / 4

A = 5

Thus, the total number of elements present in the ordered set A is 5.

• Consider an ordered pair of sets (x+9,y-9) = (3,6). Find the value of the variables.

Given that, 

The ordered pair of sets are  

(x+9,y-9) 

(3,6)

For an ordered set, according to the property of equality, equate the respective coordinates of the ordered sets.

x+9 = 3

x = 3-9

x = -6

y-9 = 6

y = 6+9

y = 15

Conclusion 

The ordered pair constitutes coordinates for plotting the graph on a Cartesian plane. When the set of ordered pairs, for example, (x+a,y+b) = (c,d), where x and y are variables and a,b,c,d are valued numbers, then applying the equality property helps us find the values of the variable.

The ordered pair definition is that the elements are present in the set in an ordered manner. Their positions cannot be changed. The Cartesian product of the ordered pair of sets gives the possible combinations of the elements of the two sets.