Given The Sets A = {1, 2, 4} And B = {0, 1, 3, 4, 5}, And A Book Costs 36 Lei. If The Price Increases By 20%, What Is The New Price Of The Book? a) Does The Pair (1,0) Belong To The Cartesian Product A X B? b) Does The Pair (1,2) Belong To The Cartesian Product A X B? c) Does The Pair (2,2) Belong To The Cartesian Product A X B? d) Does The Pair (5,4) Belong To The Cartesian Product A X B? e) What Is The New Price Of The Book?
In the realm of mathematics, set theory forms a foundational pillar upon which many other concepts are built. Understanding sets and their operations is crucial for navigating various mathematical disciplines, including algebra, calculus, and discrete mathematics. One fundamental operation involving sets is the Cartesian product, which allows us to combine elements from different sets in a structured way. In this article, we will delve into the concept of the Cartesian product, explore its properties, and illustrate its application through a concrete example. Furthermore, we will address a practical problem involving percentage increases, demonstrating the relevance of mathematical concepts in everyday scenarios.
The Cartesian product of two sets, denoted as A × B, is the set of all possible ordered pairs (a, b), where 'a' is an element of set A and 'b' is an element of set B. Each element in the Cartesian product is an ordered pair, meaning the order in which the elements are written matters. For instance, (1, 0) is a different ordered pair than (0, 1). The Cartesian product provides a systematic way to combine elements from two sets, creating a new set that captures all possible pairings. Understanding the Cartesian product is essential for comprehending relations, functions, and other advanced mathematical concepts. It serves as a building block for defining relationships between sets and mapping elements from one set to another. In various fields, such as computer science and data analysis, the Cartesian product finds applications in areas like database design and data mining, where combining data from different sources is a common requirement.
Cartesian Product Explained
Let's consider the sets A = {1, 2, 4} and B = {0, 1, 3, 4, 5}. To find the Cartesian product A × B, we systematically pair each element of A with every element of B. This process generates a set of ordered pairs, where the first element of each pair comes from A and the second element comes from B. The number of elements in the Cartesian product is equal to the product of the number of elements in the original sets. In this case, A has 3 elements and B has 5 elements, so A × B will have 3 * 5 = 15 elements. Each of these elements is an ordered pair, representing a unique combination of elements from A and B. The concept of the Cartesian product extends to more than two sets as well. For example, the Cartesian product of three sets, A, B, and C, denoted as A × B × C, consists of ordered triples (a, b, c), where 'a' belongs to A, 'b' belongs to B, and 'c' belongs to C. This extension allows us to model more complex relationships and combinations, further enhancing the applicability of the Cartesian product in various mathematical and computational contexts. The Cartesian product is a cornerstone of set theory, providing a structured way to combine elements from different sets and enabling the definition of complex relationships and mappings.
Identifying Elements in the Cartesian Product
Given the sets A = {1, 2, 4} and B = {0, 1, 3, 4, 5}, we can determine which of the following pairs belongs to the Cartesian product A × B: (1, 0), (1, 2), (2, 2), (5, 4). Recall that A × B consists of all ordered pairs (a, b) where a ∈ A and b ∈ B. Let's examine each option:
- (1, 0): 1 ∈ A and 0 ∈ B, so (1, 0) ∈ A × B.
- (1, 2): 1 ∈ A but 2 ∉ B, so (1, 2) ∉ A × B.
- (2, 2): 2 ∈ A but 2 ∉ B, so (2, 2) ∉ A × B.
- (5, 4): 5 ∉ A but 4 ∈ B, so (5, 4) ∉ A × B.
Therefore, only the pair (1, 0) belongs to the Cartesian product A × B. This exercise illustrates the importance of verifying that each element in an ordered pair comes from the correct set when determining membership in a Cartesian product. The Cartesian product provides a systematic way to generate all possible pairings between elements of two sets, but it's crucial to ensure that each element in the resulting ordered pairs belongs to its respective set. This meticulous verification ensures the integrity of the Cartesian product and its subsequent applications in various mathematical contexts. The ability to correctly identify elements within a Cartesian product is a fundamental skill in set theory, enabling us to work with relations, functions, and other advanced concepts that rely on the Cartesian product as a building block.
Percentage increases are a common concept in everyday life, often encountered in financial transactions, economic analysis, and even simple retail scenarios. Understanding how to calculate percentage increases is crucial for making informed decisions and interpreting numerical data accurately. In this section, we will explore the concept of percentage increases and apply it to a practical problem involving the price of a book. The concept of percentage increase is rooted in the idea of expressing a change in value relative to the original value. It allows us to quantify the magnitude of the change in a standardized way, making it easier to compare changes across different contexts. For instance, a 20% increase in the price of a book has the same relative impact as a 20% increase in an investment, even though the absolute amounts may differ significantly. This standardization makes percentage increases a powerful tool for analysis and decision-making.
Price Increase Problem
Let's consider the scenario where a book initially costs 36 lei, and its price is increased by 20%. To determine the new price of the book, we need to calculate the amount of the increase and add it to the original price. The first step is to find 20% of 36 lei, which represents the increase in price. To do this, we multiply 36 by 20/100, which gives us 7.2 lei. This means the price of the book increased by 7.2 lei. Next, we add this increase to the original price: 36 lei + 7.2 lei = 43.2 lei. Therefore, the new price of the book after the 20% increase is 43.2 lei. This problem illustrates a common application of percentage increases in retail and pricing contexts. Businesses often use percentage increases to adjust prices based on factors such as inflation, demand, or cost of goods. Understanding how to calculate these increases allows consumers to make informed purchasing decisions and evaluate the impact of price changes on their budgets. Furthermore, the concept of percentage increase extends beyond pricing scenarios and finds applications in various fields, including finance, economics, and statistics. In these contexts, it is used to analyze growth rates, investment returns, and other numerical trends. The ability to calculate percentage increases accurately is a valuable skill for navigating both personal and professional financial matters.
Step-by-Step Calculation
To calculate a percentage increase, we follow a straightforward process:
- Calculate the amount of the increase: Multiply the original value by the percentage increase (expressed as a decimal). In this case, 20% of 36 lei is (20/100) * 36 = 7.2 lei.
- Add the increase to the original value: This gives us the new value. In this case, 36 lei + 7.2 lei = 43.2 lei.
Therefore, the new price of the book after the 20% increase is 43.2 lei. This step-by-step approach provides a clear and concise method for calculating percentage increases in any context. By breaking down the calculation into smaller steps, we can minimize the risk of errors and ensure accurate results. This method is applicable to a wide range of scenarios, from calculating price increases in retail to determining investment returns in finance. The key is to first identify the original value and the percentage increase, and then apply the formula consistently. Understanding the underlying logic of percentage increases allows us to adapt this method to more complex situations, such as calculating multiple percentage changes or working with percentage decreases. The ability to calculate percentage increases accurately is a fundamental skill for anyone who deals with numerical data, whether in personal finance, business, or academic research.
In this article, we have explored two fundamental mathematical concepts: the Cartesian product and percentage increases. The Cartesian product provides a structured way to combine elements from different sets, forming the basis for many advanced mathematical concepts. We demonstrated how to identify elements within a Cartesian product through a concrete example. Understanding the Cartesian product is crucial for comprehending relations, functions, and other mathematical structures that rely on the systematic pairing of elements. Its applications extend beyond pure mathematics, finding relevance in fields such as computer science and data analysis, where combining data from different sources is a common requirement.
Furthermore, we addressed the practical problem of calculating percentage increases, illustrating its relevance in everyday scenarios. We demonstrated how to calculate the new price of a book after a 20% increase, highlighting the step-by-step process involved. Percentage increases are a common concept in financial transactions, economic analysis, and retail settings, making it essential to understand how to calculate them accurately. The ability to calculate percentage increases empowers individuals to make informed decisions, evaluate the impact of price changes, and analyze numerical trends in various contexts. By mastering these two concepts, the Cartesian product and percentage increases, we gain valuable tools for navigating both theoretical and practical mathematical challenges. These concepts serve as building blocks for more advanced mathematical topics and enhance our ability to interpret and analyze numerical data in everyday life.