What Is an Ordered Pair? Definition, Properties & Key Examples
An ordered pair consists of two numbers that are written in a fixed order. So, we can define an ordered pair as the pair of elements that occur in a particular order and are enclosed in brackets.
An ordered pair is a combination of the x coordinate and the y coordinate, having two values written in a fixed order within parentheses.
Overview of an Ordered Pair
An ordered pair refers to a number written in a certain order. An ordered pair is used to show the position on a graph, where the "x" (horizontal) value is first, and the "y" (vertical) value is second. Also in the co-ordinate system, ordered pair is used to locate a point. Pairs in math are denoted by (,) and are generally considered to be ordered.
Ordered Pair explanation
Ordered Pair = (x, y)
where, x = abscissa, the distance measure of a point from the x-axis.
and, y = ordinate, the distance measure of a point from the y-axis.
To graph a point, we have to draw a dot at the coordinates that correspond to the ordered pair. The x-coordinate tells us how many steps we have to take to on the x-axis. The y-coordinate tells us to have many steps to move on the y-axis.
Example of an Ordered Pair
The ordered pair (2, 5) means a pair of two integers, strictly in the order with 2 at the first place called the abscissa and 5 at the second place called the ordinate.
The ordered pair (2, 5) is not equal to the ordered pair (5, 2) because (2, 5) ≠ (5, 2). Therefore, in a pair, the order of elements is important.
Set of Ordered Pairs
The pair of elements that occur in a particular order and are enclosed in brackets is known as a set of ordered pairs. If ‘a’ and ‘b’ are two elements, then the two different pairs are (a, b), (b, a). In an ordered pair (a, b), a is called the first element, and b is called the second element.
Consider, if A and B are two sets such that a ∈ A and b ∈ B, then by the ordered pair of elements we mean (a, b) where 'a' is called the Ist element and 'b' is called the IInd element of the ordered pair.
If we change the position of the elements in the ordered pair, then the ordered pair also get changed, i.e., it becomes (b, a) but (a, b) ≠ (b, a).
Properties of Ordered Pairs
Equality of Ordered Pairs
Two ordered pairs are said to be equal if and only if the corresponding first elements are equal and the corresponding second elements are equal.
For example: Consider two ordered pairs (a, b) and (c, d). They are equal if a = c and b = d, i.e., (a, b) = (c, d).
Question: Find the values of x and y, if (2x - 3, y + 1) = (x + 5, 7)
Solution: We will solve by equality of ordered pairs
Given 2x - 3 = x + 5 and y + 1 = 7
⇒ 2x - x = 5 + 3
⇒ x = 8
and y = 7 - 1
⇒ y = 6
Hence x = 8 and y = 6.
Note: Both the elements of an ordered pair can be the same i.e., (2, 2), (5, 5).
Solved Examples
1. Plot the point “P” with coordinates 6, 4.
Sol: As per the definition of ordered pair, the point P will be written as:
P = (6, 4)
The first number in the ordered pair shows the distance from the “x" axis which is 6.
The second number in the ordered pair shows the distance from the “y" axis which is 4.
To locate the point on the Cartesian plane, start from the origin. Take 6 steps towards the “x” axis (towards the right) starting from the origin. From here, move 4 steps towards the “y” axis (upwards).
As we know the order in which values are written in an ordered pair is very important. The ordered pair (6, 4) is different from the ordered pair (4, 6). Both represent two different points as shown below.
2. If (x, y) = (-1, 4) find the value of 3x + 2y - 4?
Sol: Given x = (-1) and y = 4. Substitute x, y value in given equation
3x + 2y - 4 = 3(-1) + 2(4) - 4 = 1.
Fun Facts
The mathematician Rene Descartes and Pierre de Fermat invented analytic geometry in the 16th century and the Cartesian plane was designed.
Ordered pair is widely used in the field of computing and programming languages.
Ordered pairs are also known as 2-tuples, or sequences of length 2.
Summary
A pair of numbers written in a particular order and enclosed in brackets are known as ordered pairs.
And ordered pair is represented as, Ordered Pair = (x, y)
The ordered pair changes based on the change in the position of the elements of the ordered pair.
It is used in statistics, to depict the center and vertices of a circle, square, rectangle, and others.
Ordered pairs are used for data comprehension and to aid visual comprehension.
FAQs on Ordered Pair in Maths: Complete Guide
1. What is an ordered pair in mathematics, with an example?
An ordered pair is a pair of mathematical elements or objects in a specific, fixed sequence. It is represented by listing the two elements within parentheses, separated by a comma. The key characteristic is that the order matters. For example, the ordered pair (3, 5) is different from the ordered pair (5, 3) because the positions of the elements are reversed. Ordered pairs are fundamental in coordinate geometry and set theory.
2. What are the two components of an ordered pair called when used in coordinate geometry?
When an ordered pair (x, y) is used to locate a point on a Cartesian plane, its components have specific names:
The first element, x, is called the x-coordinate or the abscissa. It represents the point's horizontal position relative to the origin.
The second element, y, is called the y-coordinate or the ordinate. It represents the point's vertical position relative to the origin.
3. How is an ordered pair like (a, b) different from a set like {a, b}?
The primary difference lies in the importance of order and the treatment of duplicate elements.
Order: In an ordered pair, the sequence is critical. For instance, (a, b) ≠ (b, a) unless a = b. In a set, the order is irrelevant; the set {a, b} is identical to {b, a}.
Repetition: An ordered pair can have identical elements, such as (2, 2). In a set, elements are unique, so a set with repeated elements like {2, 2} is simply written as {2}.
4. What is the importance of ordered pairs in defining the Cartesian Product of sets?
Ordered pairs are the fundamental building blocks of the Cartesian Product. The Cartesian Product of two non-empty sets, A and B, denoted as A × B, is the set of all possible ordered pairs (a, b) where the first element 'a' is from set A and the second element 'b' is from set B. For example, if A = {1, 2} and B = {p, q}, then A × B = {(1, p), (1, q), (2, p), (2, q)}. Without the concept of ordered pairs, this structured combination of elements would not be possible.
5. Why is the order of elements so crucial in an ordered pair?
The order is crucial because it assigns a specific role or meaning to each element, which is essential for representing relationships and locations. For example:
In coordinate geometry, (4, 7) represents a unique point on a plane. The point (7, 4) is a completely different location.
In real-world data, if an ordered pair represents (age, weight), then (25, 70) means 25 years old and 70 kg. Reversing it to (70, 25) would completely change its meaning. The order provides the context.
6. How do ordered pairs form the basis for Relations and Functions in mathematics?
In set theory, a relation between two sets is defined as any subset of their Cartesian product. This means a relation is simply a set of ordered pairs that follow a specific rule. A function is an even more specific type of relation where each first element of an ordered pair is associated with exactly one second element. Therefore, the ordered pair (input, output) is the fundamental structure used to define and understand both relations and functions.
7. When are two ordered pairs considered equal?
Two ordered pairs, (a, b) and (c, d), are considered equal if and only if their corresponding elements are equal. This means that for (a, b) = (c, d), two conditions must be met simultaneously: a = c (the first elements must be equal) and b = d (the second elements must be equal). If either of these conditions is not met, the ordered pairs are not equal.
