Expressing the Set of Ordered Pairs in Set Builder Notation

The process of expressing a set in set builder notation involves identifying a pattern and specifying the conditions that define the elements of the set clearly. This article will explore how to express the set of ordered pairs in set builder notation, particularly when the elements of one component are related to the other by a specific relationship.

Identifying the Pattern

In the given set {00, 12, 36, 48, ...}, we need to identify the pattern that connects the first and second elements of each ordered pair. By examining the pairs:

(0, 0) (1, 2) (3, 6) (4, 8)

We observe that the first element, x, and the second element, y, follow a specific relationship. In the simplest case:

For x 0, y 0 For x 1, y 2 For x 3, y 6 For x 4, y 8

It becomes clear that for each pair, the second element, y, is twice the first element, x. That is:

y 2x

Expressing the Set in Set Builder Notation

Once the relationship is identified, we need to express it in set builder notation. Set builder notation typically follows the format:

{ (x, y) | x ∈ S and y f(x) }

Here, (x, y) represents the ordered pair, S is the set defining x, and f(x) is the function that defines y in terms of x.

From the given data:

The x values are 0, 1, 3, 4. The y values are 0, 2, 6, 8, and follow the pattern y 2x.

We can express this set as:

{ (x, y) | x ∈ {0, 1, 3, 4} and y 2x }

Extending to a Broader Set of Natural Numbers

While the provided set is explicit, if the pattern continues indefinitely, we can extend this to all natural numbers, including zero. The broader notation for the set of all pairs (x, y) such that y is twice x, for all natural numbers, becomes:

{ (x, y) | x ∈ ?, y 2x }

Where ? represents the set of all natural numbers, which includes 0, 1, 2, 3, etc.

Conclusion

Understanding the underlying pattern and expressing it in set builder notation is a powerful method to describe sets of ordered pairs. This notation not only clarifies the relationship between elements but also provides a clear and concise way to represent complex sets.

Related Keywords

- Set Builder Notation

- Ordered Pairs

- Pattern Recognition