An ordered pair, as the name suggests, is a pair of elements that have specific importance to the order of their locations. Ordered pairs are generally used in coordinate geometry to represent a point on a coordinate plane. Also, they are used to represent elements of a relationship.
Let’s learn more about ordered pairs along with their definition, meaning, properties, and more.
1. What is an ordered pair? 2. Ordered Pair in Coordinate Geometry 3. Graphing Ordered Pairs 4. Ordered Pairs in Different Quadrants 5. Ordered Pair in Sets 6. Ordered Pair Property of Equality 7. Ordered Pair FAQ
An ordered pair is a pair consisting of two elements that are separated by a comma and enclosed in parentheses. For example, (x, y) represents an ordered pair, where ‘x’ is called the first element and ‘y’ is called the second element of the ordered pair. These elements have specific names depending on the context in which they are used and can be variables or constants. The order of the elements has some importance in an ordered pair. It means that (x, y) may not be equal to (y, x) all the time.
(2, 5), (a, b), (0, -5), etc. are some examples of ordered pairs.
An ordered pair in coordinate geometry is used to represent the position of a point in the coordinate plane with respect to the origin. A coordinate plane is formed by two perpendicular lines that intersect between which one is horizontal (x axis) and the other line is vertical (y axis). The intersection point of both axes is the origin. Each point in the coordinate plane is represented by an ordered pair (x, y) where the first element x is called the x-coordinate and the second element y is called the y-coordinate. We can see more differences between the elements of the ordered pair used in geometry here.
First Element of the Ordered Pair Second Element of the Ordered Pair It is called the x-coordinate. It is called the y-coordinate. Another name for this is “abscissa”. Another name for this is “ordered”. Represents the horizontal distance of the point from the origin. Represents the vertical distance of the point from the origin. This number is one of the numbers on the x-axis. This number is one of the numbers on the y axis. Represents the distance of the point from the y axis. Represents the distance of the point from the x-axis. Example: If (2, 4) is a point on the coordinate plane, then 2 is the point’s distance from the y-axis.
Example: If (2, 4) is a point on the coordinate plane, so 2 is the distance of the point from the y-axis.
Now, we understand the difference between the x-coordinate and the y-coordinate of an ordered pair in coordinate geometry. Let’s now look at the steps for graphing ordered pairs.
- Step 1: Always start at the origin and move horizontally along |x| units to the right if x is positive and to the left if x is negative. Stay there.
- Step 2: Start from where you stopped in Step 1 and move vertically across |y| units up if y is positive and down if y is negative. Stay there.
- Step 3: Place a point exactly where you stopped in Step 2 and that point represents the ordered pair (x , y)
In these steps, |x| and |and| represent the absolute values of x and y respectively.
Example: Graph the ordered pair (4, -3).
Let’s start from the origin, move to the right 4 units (since 4 is positive) and then down 3 units (since 3 is negative).
The order of the elements in an ordered pair is important, hence the name “ordered” pair. For example, (4, -3) and (-3, 4) are located at different positions in the plane, as shown below.
We can see in the previous figure that the coordinate plane is divided into 4 parts by the x and y axes. Each of these 4 parts is known as a quadrant. The signs of x and y in an ordered pair (x, y) of a point differ depending on the quadrant and are shown in the following table.
Signs of ordered pairs of quadrant I xSo (2, -4) is a point in quadrant IV.
So far we have seen that ordered pairs are used in coordinate geometry to locate a point. But they are also used in set theory in a different context. The set of all possible ordered pairs from set A to set B is called the Cartesian product. For example, if A = {1, 2, 3} and B = {a, b, c}, then the Cartesian product is A x B = {(1, a), (1, b), (1, c ), (2, a), (2, b), (2, c), (3, a), (3, b), (3, c)} and is a set of all ordered pairs (x , y) where x is in A and y is in B. Any subset of the Cartesian product is called a relation. For example, {(1, a), (1, b), (3, c)} is a relation.
Examples:
- If (2, 4) belongs to a “divides” relation, it means that 2 divides 4.
- If (4, 2) belongs to a “greater than” relation, it means that 4 is greater that 2.
- If (x, y) belongs to a relation “is sister of”, it means that x is sister of y.
Because any two ordered pairs (x, y) and (a, b) (either in coordinate geometry or in relations), if (x, y) = (a, b) then x = a and y = b. that is, if two ordered pairs are equal, then their corresponding elements are equal. This is called the “ordered pairs property of equality”. For example:
- If (x, y) = (2, -3) then x = 2 and y = -3.
- If (x 1, y – 2 ) = (-3, 5) so x 1 = -3 and y – 2 = 5.
Important notes about ordered pairs:
- An ordered pair (x, y) is used to represent the location of a point in coordinate geometry where x is the horizontal distance and y is the vertical distance of the point.
- An ordered pair (x, y) represents an element of a relation R which is denoted by xRy (x “is related to” y).
- If (x, y) = (a, b) then x = a and y = b.
Related Topics:
- Ordered Pairs Calculator
- Introduction to Graphing
- Graphing functions
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