Question

Given \(S_{1}=\\{3,6,9\\}, S_{2}=\\{a, b\\},\) and \(S_{3}=\\{m, m\\},\) find the Cartesian products: (a) \(S_{1} \times S_{2}\) \((b) S_{2} \times S_{3}\) (c) \(S_{3} \times S_{1}\)

Short answer

(a) \(S_{1} \times S_{2} = \{(3,a),(3,b),(6,a),(6,b),(9,a),(9,b)\}\) (b) \(S_{2} \times S_{3} = \{(a,m),(a,m),(b,m),(b,m)\}\) (c) \(S_{3} \times S_{1} = \{(m,3),(m,6),(m,9),(m,3),(m,6),(m,9)\}\)

Step-by-step solution

Understanding Cartesian Product

The Cartesian product of two sets is the set of all ordered pairs where the first element comes from the first set and the second element comes from the second set. If you have two sets A and B, their Cartesian product is denoted as \(A \times B\). It consists of all combinations \((a, b)\) where \(a \in A\) and \(b \in B\).

Cartesian Product of S1 and S2

For solving part (a), we calculate \(S_{1} \times S_{2}\). We pair each element of \(S_{1}\) with every element of \(S_{2}\). The pairs are (3, a), (3, b), (6, a), (6, b), (9, a), and (9, b).

Cartesian Product of S2 and S3

For part (b), we calculate \(S_{2} \times S_{3}\). Here, \(S_{3}\) has two indistinguishable elements 'm', but the Cartesian product considers them as separate elements. Thus, the pairs are (a, m), (a, m), (b, m), and (b, m).

Cartesian Product of S3 and S1

For part (c), we need \(S_{3} \times S_{1}\). Again, every element from \(S_{3} = \{m, m\}\) is paired with every element of \(S_{1}\). The resulting pairs are (m, 3), (m, 6), (m, 9), (m, 3), (m, 6), and (m, 9).

Key concepts

Understanding Ordered Pairs

In set theory, an ordered pair is a collection of two elements where the order in which these elements appear is significant. This is unlike plain sets, where order doesn't matter.
For example, the ordered pair (3, a) is different from (a, 3), as the first refers to an element 3 coming from one set and an element a from another set.
In the context of Cartesian products, the ordered pair structure is crucial. It helps in defining the relationship between two different sets. For instance, in the exercise problem given, we form ordered pairs by aligning the first item from one set and the second item from another.
This makes ordered pairs foundational in expressing relationships and functionalities among sets in mathematics.

Exploring Set Theory

Set theory is the mathematical study of collections of items or objects, which are referred to as 'sets.'
Sets are fundamental because they are used to build other mathematical structures.

• Element of a Set: A member or an item in a set. For instance, in the set \(S_{1} = \{3, 6, 9\}\), 3 is an element.
• Subset: A smaller set defined from a larger one, containing elements only from the original set.

In the Cartesian product concept, the theory shows its strengths by laying out how elements from different sets interact to create a new set of ordered pairs. Set theory provides the structure and language necessary to discuss these relationships effectively.

Delving into Mathematical Operations

Mathematical operations allow us to perform various processes on numbers or sets, giving us valuable insights or results.
The Cartesian product is one such operation in the context of sets. It 'operates' on sets to create ordered pairs. It's similar to multiplication but instead of numbers, we're working with sets and their elements. The result is a new set of all possible ordered pairs. Another key operation is determining relations between two sets. The Cartesian product helps in illustrating these relations explicitly through ordered pairs. Understanding this operation aids in handling more complex math goals and functions that require set manipulation.
So, the Cartesian product is a critical operation in math that combines elements from multiple sets to explore possible connections or mappings.

Finite Sets in Mathematics

A finite set is a set with a specific, countable number of elements. These are sets where you can list all elements without coming to infinite stretch.
In the exercise you're reviewing, the sets \(S_{1}\), \(S_{2}\), and \(S_{3}\) are finite. Each set contains a limited number of distinct items, allowing us to easily list all members and generate ordered pairs in Cartesian products.

• Importance in Cartesian Products: Finite sets keep the Cartesian product manageable. Non-finite sets could result in overwhelming and uncountable ordered pairs.

Working with finite sets simplifies the exploration of relationships between sets as it turns the process into a manageable and calculable task.