S = {(x, Y) : X ∈ A, Y ∈ A} A = {x ∈ Z : X² ≤ 9}

Introduction to Set Theory and the Definition of S

In the realm of mathematics, set theory forms a fundamental cornerstone upon which many other concepts are built. Sets, at their core, are collections of distinct objects, which can range from numbers and symbols to even other sets themselves. Understanding the notation and operations related to sets is crucial for grasping more advanced mathematical topics. In this article, we will delve into a specific set, S, defined as S = (x, y) x ∈ A, y ∈ A, where A is another set defined as A = x ∈ Z x² ≤ 9. To fully comprehend the set S, we must first dissect the definition of set A, understand the implications of the Cartesian product, and then explore the elements that constitute S.

Set A is defined as the set of all integers (denoted by Z) such that the square of each integer is less than or equal to 9. Mathematically, this is represented as A = x ∈ Z x² ≤ 9. This notation is a concise way of specifying a set based on a particular condition or property. To decipher this, we need to identify all integers whose squares do not exceed 9. Let's consider some integers and their squares:

• (-4)² = 16
• (-3)² = 9
• (-2)² = 4
• (-1)² = 1
• (0)² = 0
• (1)² = 1
• (2)² = 4
• (3)² = 9
• (4)² = 16

From this, we can observe that the integers -3, -2, -1, 0, 1, 2, and 3 satisfy the condition x² ≤ 9. Therefore, set A can be explicitly written as A = {-3, -2, -1, 0, 1, 2, 3}. This set forms the building block for understanding the set S.

Now, let's turn our attention to the definition of set S. It is defined as S = (x, y) x ∈ A, y ∈ A. This notation describes a set of ordered pairs (x, y), where both x and y are elements of set A. The set S is essentially the Cartesian product of set A with itself, denoted as A × A. The Cartesian product of two sets is the set of all possible ordered pairs where the first element comes from the first set and the second element comes from the second set. In this case, since both sets are A, we are considering all possible pairs of elements from A.

To illustrate this further, consider the elements of A: {-3, -2, -1, 0, 1, 2, 3}. To form the set S, we pair each element of A with every other element of A, including itself. This process results in the following ordered pairs:

• (-3, -3), (-3, -2), (-3, -1), (-3, 0), (-3, 1), (-3, 2), (-3, 3)
• (-2, -3), (-2, -2), (-2, -1), (-2, 0), (-2, 1), (-2, 2), (-2, 3)
• (-1, -3), (-1, -2), (-1, -1), (-1, 0), (-1, 1), (-1, 2), (-1, 3)
• (0, -3), (0, -2), (0, -1), (0, 0), (0, 1), (0, 2), (0, 3)
• (1, -3), (1, -2), (1, -1), (1, 0), (1, 1), (1, 2), (1, 3)
• (2, -3), (2, -2), (2, -1), (2, 0), (2, 1), (2, 2), (2, 3)
• (3, -3), (3, -2), (3, -1), (3, 0), (3, 1), (3, 2), (3, 3)

Each row represents the pairs formed by fixing the first element and varying the second element across all members of A. In total, there are 7 * 7 = 49 such pairs. Therefore, the set S contains 49 ordered pairs. Understanding the composition of S is crucial for various mathematical applications, including relations, functions, and graphical representations.

Delving Deeper into Set A: Integers and Inequalities

Set A, defined as A = x ∈ Z x² ≤ 9, is the cornerstone for understanding the set S. To fully grasp the nature of S, we must first thoroughly analyze the elements that constitute A. The definition of A involves two key components: the set of integers (Z) and an inequality (x² ≤ 9). The set of integers, denoted by Z, includes all whole numbers and their negatives, extending infinitely in both positive and negative directions (..., -3, -2, -1, 0, 1, 2, 3, ...). The inequality x² ≤ 9 imposes a condition that restricts the elements of A to only those integers whose squares are less than or equal to 9.

To identify the integers that satisfy the inequality x² ≤ 9, we can consider the squares of various integers and check if they meet the condition. Alternatively, we can approach this algebraically. The inequality x² ≤ 9 can be rewritten as -3 ≤ x ≤ 3. This is because the square root of 9 is 3, and we must consider both positive and negative roots since squaring a negative number results in a positive number. Therefore, any integer x that falls within the range of -3 to 3, inclusive, will satisfy the inequality.

Let's examine the integers within this range: -3, -2, -1, 0, 1, 2, and 3. Squaring each of these integers, we get:

• (-3)² = 9
• (-2)² = 4
• (-1)² = 1
• (0)² = 0
• (1)² = 1
• (2)² = 4
• (3)² = 9

As we can see, all these squares are indeed less than or equal to 9. Any integer outside this range will have a square greater than 9. For example, (-4)² = 16 and (4)² = 16, both of which exceed 9. Therefore, the set A consists precisely of the integers -3, -2, -1, 0, 1, 2, and 3. We can write this explicitly as A = {-3, -2, -1, 0, 1, 2, 3}.

The significance of set A lies in its role as the domain and range for the ordered pairs that constitute set S. Since S is defined as S = (x, y) x ∈ A, y ∈ A, both the first element (x) and the second element (y) of each ordered pair must be members of A. This constraint limits the possible combinations of pairs, making A a fundamental building block for understanding S. The cardinality (number of elements) of A, which is 7 in this case, directly impacts the cardinality of S. The set A is a finite set, meaning it has a countable number of elements, which simplifies the analysis of S.

Understanding the nature of set A also helps in visualizing S. If we consider a coordinate plane, the elements of S can be represented as points. Since both x and y coordinates are limited to the integers in A, the points will be discrete and confined to a specific region. This discrete nature is a direct consequence of the integer constraint imposed on A. The inequality x² ≤ 9 plays a crucial role in determining the boundaries of this region. The interplay between the integer constraint and the inequality shapes the characteristics of both set A and, consequently, set S. Therefore, a thorough understanding of A is paramount for a comprehensive grasp of S.

Constructing Set S: The Cartesian Product A × A

The set S, defined as S = (x, y) x ∈ A, y ∈ A, is formed by taking the Cartesian product of set A with itself. Understanding the Cartesian product is crucial to grasping the composition and characteristics of S. The Cartesian product of two sets, say A and B, denoted as A × B, is the set of all possible ordered pairs (a, b), where 'a' is an element of A and 'b' is an element of B. In the case of set S, we are considering A × A, which means both elements of the ordered pairs are drawn from the same set, A.

Given that A = {-3, -2, -1, 0, 1, 2, 3}, we can systematically construct the set S by pairing each element of A with every other element of A, including itself. This process ensures that we capture all possible combinations, adhering to the definition of the Cartesian product. To illustrate this, let's consider the first element of A, which is -3. We can pair -3 with each element of A to form the following ordered pairs: (-3, -3), (-3, -2), (-3, -1), (-3, 0), (-3, 1), (-3, 2), and (-3, 3). These pairs represent the first row of elements in S.

We then repeat this process for the second element of A, which is -2, resulting in the following pairs: (-2, -3), (-2, -2), (-2, -1), (-2, 0), (-2, 1), (-2, 2), and (-2, 3). This forms the second row of elements in S. We continue this pattern for each element of A, systematically generating all possible ordered pairs. The resulting set S consists of all these pairs:

• S = {(-3, -3), (-3, -2), (-3, -1), (-3, 0), (-3, 1), (-3, 2), (-3, 3),
• (-2, -3), (-2, -2), (-2, -1), (-2, 0), (-2, 1), (-2, 2), (-2, 3),
• (-1, -3), (-1, -2), (-1, -1), (-1, 0), (-1, 1), (-1, 2), (-1, 3),
• (0, -3), (0, -2), (0, -1), (0, 0), (0, 1), (0, 2), (0, 3),
• (1, -3), (1, -2), (1, -1), (1, 0), (1, 1), (1, 2), (1, 3),
• (2, -3), (2, -2), (2, -1), (2, 0), (2, 1), (2, 2), (2, 3),
• (3, -3), (3, -2), (3, -1), (3, 0), (3, 1), (3, 2), (3, 3)}

The cardinality (number of elements) of S can be determined by multiplying the cardinality of A by itself. Since A has 7 elements, the cardinality of S is 7 * 7 = 49. This means that S consists of 49 distinct ordered pairs. The ordered nature of the pairs in S is significant. The pair (x, y) is distinct from the pair (y, x) unless x = y. This distinction is a fundamental characteristic of ordered pairs and the Cartesian product.

The set S, formed through the Cartesian product, has numerous applications in mathematics. It can represent relations between elements of A, where each pair (x, y) signifies a relationship between x and y. For instance, if we define a relation R such that (x, y) ∈ R if x < y, then we can identify the pairs in S that satisfy this condition. Furthermore, S can be visualized as a grid of points in a two-dimensional coordinate system, where the x and y coordinates are restricted to the integers in A. This geometric interpretation provides a visual aid for understanding the structure and properties of S.

The Cartesian product also plays a vital role in defining functions. A function from A to A can be seen as a subset of S, where each element in A is mapped to a unique element in A. Understanding S as the Cartesian product A × A allows for a systematic exploration of possible functions and relations that can be defined on the set A. Therefore, the construction of set S through the Cartesian product is not just a mechanical process but a foundational step towards exploring deeper mathematical concepts.

Applications and Implications of S in Mathematics

The set S = (x, y) x ∈ A, y ∈ A, where A = x ∈ Z x² ≤ 9, serves as a foundational structure for understanding various mathematical concepts and has significant applications in different areas. Its implications extend to relations, functions, graph theory, and even basic programming concepts. The ordered pairs in S represent connections or mappings between elements of A, making it a versatile tool for mathematical modeling and analysis.

One of the primary applications of S lies in the definition and representation of relations. A relation from a set A to itself is a subset of the Cartesian product A × A. Since S is defined as A × A, any subset of S can be considered a relation on A. For example, we can define a relation R on A such that (x, y) ∈ R if x is less than y. To explicitly list the elements of R, we would examine the pairs in S and select those that satisfy the condition x < y. This would include pairs like (-3, -2), (-1, 2), and (0, 3), among others. The relation R captures a specific type of association between elements of A, in this case, the