{a, b} × {1, 2, 3}

Definition

The expression {a, b} × {1, 2, 3} represents the Cartesian product of two sets, {a, b} and {1, 2, 3}. This means that it creates a new set containing all possible ordered pairs where the first element comes from the first set and the second element comes from the second set. The Cartesian product is fundamental in understanding ordered pairs and their role in mathematical relations and functions.

5 Must Know Facts For Your Next Test

1. The Cartesian product {a, b} × {1, 2, 3} results in the set {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}.
2. The number of elements in the Cartesian product of two sets is equal to the product of the number of elements in each set. In this case, it is 2 × 3 = 6.
3. Cartesian products can be visualized as points in a grid where each axis corresponds to one of the sets involved.
4. The concept of Cartesian products is widely used in various fields such as mathematics, computer science, and statistics for defining relations and functions.
5. The order of the sets matters; {a, b} × {1, 2} is not the same as {1, 2} × {a, b}, resulting in different sets of ordered pairs.

Review Questions

• What is the result of the Cartesian product {a, b} × {1, 2, 3}, and how does this relate to ordered pairs?
  • The Cartesian product {a, b} × {1, 2, 3} produces the set {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}. This relates to ordered pairs because each element of this new set is an ordered pair formed by taking one element from each original set. The first element in each pair comes from the first set and the second element from the second set.
• Explain how the Cartesian product demonstrates the relationship between two sets and give an example using different sets.
  • The Cartesian product illustrates how elements from two distinct sets can be combined to form pairs that represent relationships between those elements. For example, if we take sets {x, y} and {4, 5}, their Cartesian product {x, y} × {4, 5} would yield {(x, 4), (x, 5), (y, 4), (y, 5)}. Each pair shows a connection between an element from the first set and an element from the second set.
• Evaluate the significance of understanding Cartesian products when discussing relations in mathematics.
  • Understanding Cartesian products is crucial for discussing relations because they form the foundational basis for defining how elements from two sets interact with each other. By recognizing how to construct these products through ordered pairs, one can explore various mathematical concepts like functions and mappings. Additionally, knowing how to manipulate these relationships helps in solving complex problems involving multi-dimensional data and structures.

Related terms

Ordered Pair:

An ordered pair is a pair of elements where the order in which the elements appear is significant, typically written as (x, y).

Set: A set is a collection of distinct objects, considered as an object in its own right. Sets are fundamental building blocks in mathematics.

Relation: A relation is a set of ordered pairs that describes a relationship between elements of two sets.