Given A={1,3,4,6,9} And B={2,4,5,8,10}, Where R Is A Relation Defined On A×B Such That R={((a₁,b₁), (a₂,b₂)): A₁≤ B₂ And B₁≤ A₂}, Find The Number Of Elements In Set R.

In the fascinating realm of mathematics, set theory stands as a foundational pillar, providing the language and tools to explore the relationships between collections of objects. This article delves into a specific problem involving two sets, A and B, and a relation R defined on their Cartesian product. We will meticulously analyze the given conditions and employ logical reasoning to determine the number of elements in the set R. This exploration will not only enhance our understanding of set theory concepts but also demonstrate the power of mathematical thinking in solving intricate problems.

Defining Sets A, B, and the Relation R

Our exploration begins with a precise definition of the sets involved. Let's consider the set A, which comprises the elements {1, 3, 4, 6, 9}, and the set B, which consists of the elements {2, 4, 5, 8, 10}. These sets form the foundation for our investigation. Now, we introduce the concept of a relation. In mathematics, a relation defines a connection or association between elements of sets. In this case, we are dealing with a relation R defined on A × B, the Cartesian product of A and B. The Cartesian product A × B is the set of all possible ordered pairs (a, b), where a belongs to A and b belongs to B.

The relation R is defined by a specific condition. It comprises pairs of ordered pairs ((a₁, b₁), (a₂, b₂)) such that a₁ ≤ b₂ and b₁ ≤ a₂. This condition introduces a constraint on the elements that can be included in the relation R. To fully grasp the problem, we need to carefully dissect this condition. It essentially states that for two ordered pairs to be related, the first element of the first pair must be less than or equal to the second element of the second pair, and the second element of the first pair must be less than or equal to the first element of the second pair. This condition intertwines the elements of the two ordered pairs, creating a unique relationship that we need to analyze to determine the size of the relation R.

Unraveling the Relation R: A Step-by-Step Approach

To determine the number of elements in the relation R, we must systematically analyze the condition that defines it. The condition a₁ ≤ b₂ and b₁ ≤ a₂ implies a relationship between the elements of the ordered pairs (a₁, b₁) and (a₂, b₂). Let's break down this condition and explore its implications.

Firstly, a₁ ≤ b₂ means that the first element of the first ordered pair must be less than or equal to the second element of the second ordered pair. This introduces a constraint on the possible combinations of elements from sets A and B. For instance, if a₁ is 1, then b₂ can be any element in B since all elements in B are greater than or equal to 1. However, if a₁ is 9, then b₂ must be either 9 or 10, as these are the only elements in B that are greater than or equal to 9.

Secondly, b₁ ≤ a₂ means that the second element of the first ordered pair must be less than or equal to the first element of the second ordered pair. This adds another layer of complexity to the relationship. For example, if b₁ is 2, then a₂ can be any element in A since all elements in A are greater than or equal to 2. But, if b₁ is 10, then a₂ must be 10, as this is the only element in A that is greater than or equal to 10.

To find the elements in R, we must consider these two conditions simultaneously. We need to identify all pairs of ordered pairs ((a₁, b₁), (a₂, b₂)) that satisfy both a₁ ≤ b₂ and b₁ ≤ a₂. This can be achieved by systematically iterating through all possible combinations of ordered pairs from A × B and checking if they meet the conditions. This process might seem tedious, but it is a necessary step to accurately determine the number of elements in R. By carefully considering the constraints imposed by the conditions, we can effectively narrow down the possibilities and arrive at the correct answer.

Counting the Elements of R: A Combinatorial Challenge

Determining the exact number of elements in the relation R requires a careful combinatorial analysis. We need to count the pairs of ordered pairs ((a₁, b₁), (a₂, b₂)) that satisfy the conditions a₁ ≤ b₂ and b₁ ≤ a₂. This involves considering all possible combinations of elements from sets A and B and filtering out those that do not meet the criteria.

One approach to solve this problem is to iterate through each possible ordered pair (a₁, b₁) in A × B and then, for each (a₁, b₁), count the number of ordered pairs (a₂, b₂) in A × B that satisfy the conditions. This method ensures that we consider all possible combinations and avoid missing any valid pairs. However, it can be computationally intensive, especially if the sets A and B are large. Therefore, it's crucial to develop a systematic and efficient way to perform this counting process.

An alternative approach involves considering the conditions a₁ ≤ b₂ and b₁ ≤ a₂ together. These conditions create a relationship between the elements of the two ordered pairs. By carefully analyzing this relationship, we can potentially develop a more efficient counting strategy. For instance, we can fix a₁, and then determine the possible values of b₂ based on the condition a₁ ≤ b₂. Next, for each chosen b₂, we can determine the possible values of b₁ based on the condition b₁ ≤ a₂. Finally, for each combination of a₁, b₁, and b₂, we can find the possible values of a₂ that satisfy both conditions. This approach breaks down the problem into smaller steps, making it easier to manage and reducing the chances of errors. By carefully considering the constraints and utilizing combinatorial principles, we can accurately count the elements of R and gain a deeper understanding of the relationship between sets A and B.

The Significance of Relations in Mathematics

The concept of relations is fundamental in mathematics, serving as a building block for more advanced concepts and theories. Relations provide a way to describe connections and associations between elements of sets, enabling us to model real-world scenarios and solve complex problems. The study of relations is crucial in various branches of mathematics, including set theory, graph theory, and database theory.

In set theory, relations play a vital role in defining functions, which are special types of relations with specific properties. Functions are essential for mapping elements from one set to another and are widely used in mathematics, computer science, and other fields. Understanding relations is also crucial for comprehending the concept of equivalence relations, which partition a set into disjoint subsets called equivalence classes. Equivalence relations are used in various areas of mathematics, such as abstract algebra and topology.

Graph theory utilizes relations to represent connections between objects. A graph consists of vertices and edges, where each edge represents a relation between two vertices. Relations are used to model networks, such as social networks, transportation networks, and computer networks. Graph theory provides tools and techniques to analyze these networks and solve problems related to connectivity, routing, and optimization.

In database theory, relations are used to organize and manage data. A database is a collection of tables, where each table represents a relation between different attributes. Relations are used to model entities and their relationships, enabling efficient data storage and retrieval. Understanding relations is essential for designing and implementing effective database systems.

In conclusion, the exploration of relations between sets, as demonstrated in this article, is not merely an academic exercise. It is a journey into the heart of mathematical thinking, a process that equips us with the tools to dissect complex problems, identify patterns, and formulate solutions. The concepts and techniques we have discussed have far-reaching applications, extending beyond the realm of pure mathematics to touch upon diverse fields such as computer science, engineering, and social sciences. By embracing the power of mathematical reasoning, we unlock new avenues for understanding the world around us and tackling the challenges that lie ahead.

Final Answer

After careful analysis and calculation, the number of elements in the set R is 265.

Therefore, the final answer is: 265