As we observed in the Introduction, mathematics is often used to model the world around us by giving a precise description of the relationships between different measurable quantities. For example, if you are riding on a Ferris wheel that is 90 ft. in diameter, with the bottom of the wheel 13 ft. off the ground:
and we let y = your height off the ground (in feet), x = your displacement off center (in feet), then (by a little bit of analytic geometry) we can show that x and y are related by the equation:
x2 + (y - 58)2 = 452
We can express this relationship numerically, by listing some of the pairs of values that satisfy this equation:
| x | y | |
| 0 0 45 -45 27 27 · · · |
103 13 58 58 94 22 · · · |
This is at the top of the wheel |
Notice that this numerical description is not as complete as the algebraic description; there are infinitely many pairs of numbers that satisfy the equation, so we have neither time nor space to include all of them (as indicated by the vertical dots).
We can also express this relationship graphically, by plotting these pairs of points (as well as all the others that we could not list) on a pair of coordinate axes:
Note: The only reason that we can make such an accurate plot, and include an infinite number of points for which we had no values, is because we recognize that is the equation for a circle; we will return to this important principle of graphing later.
This is a typical example of a relation. Formally:
A relation is a collection of ordered pairs.
This mathematical concept is very flexible, and can be used to express many types of relationships. For example, the relation "is a sibling of" in my mother's family would consist of the set of ordered pairs: {(Isabella, John), (John, Isabella), (Isabella, Henry), (Henry, Isabella), (Henry, John), (John, Henry)}. Note: Because these are ordered pairs, the order matters, so that (since the relationship of sibling is reciprocal) we must list both ordered pairs for each pair of siblings. We can represent this as a table:
|
"is a sibling of" |
|
| Sibling 1 | Sibling 2 |
| Isabella John Isabella Henry Henry John |
John Isabella Henry Isabella John Henry |
or with a simple arrow diagram:
Two, more mathematical examples are:
|
"y is the square of x" |
"y is a square root of x" |
|||
| x | y | x | y | |
| -3 -2 -1 0 1 2 3 · · · |
9 4 1 0 1 4 9 · · · |
and | 9 4 1 0 1 4 9 · · · |
-3 -2 -1 0 1 2 3 · · · |
Important: While we often associate variables with the values of a relation, these are only for convenience and an not an essential part of the relation; that is,
|
"t is the square of s" |
"a is a square root of b" |
|||
| s | t | b | a | |
| -3 -2 -1 0 1 2 3 · · · |
9 4 1 0 1 4 9 · · · |
and | 9 4 1 0 1 4 9 · · · |
-3 -2 -1 0 1 2 3 · · · |
are exactly the same relations as the two given above, since a relation is determined by its pairs of values, and not any notation we may use to describe it.
These relations happen to be reverses of one another, which, in this context, means that the order of the pairs are simply reversed. We will discuss this concept in more detail, once we have explored the function concept, and we will see that this simple idea is really the foundation of most of Algebra.
For now, try the following exercises to solidify your understanding of relations, before moving on to the next section. These exercises will help you to learn to change your "point of view" on a given relation: from an equation, to a set of ordered pairs, to a table of values, to an arrow diagram, to a plot in the plane.
Make sure that you understand these concepts by completing the following exercises.
Go to Introduction to Functions
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