Question

Do you think the statement \((\mathbb{R}-\mathbb{Z}) \times \mathbb{N}=(\mathbb{R} \times \mathbb{N})-(\mathbb{Z} \times \mathbb{N})\) is true, or false? Justify.

Short answer

The statement is true, because both sides of the equation describe the same set, namely all ordered pairs where the first element is a non-integer real number and the second element is a natural number.

Step-by-step solution

Understanding the Symbols

Firstly it's important to understand that \(\mathbb{R}\) represents the set of all Real numbers, \(\mathbb{Z}\) represents the set of all Integers and \(\mathbb{N}\) represents the set of all Natural numbers. The '-' sign represents set subtraction, and \(\times\) represents a Cartesian product, which is the product of two sets.

Analyzing the Left side of the Equation

The set \((\mathbb{R}-\mathbb{Z})\) will encompass all real numbers excluding integers. When Cartesian product with \(\mathbb{N}\) (set of natural numbers) is taken, the result is the set of all ordered pairs, where the first element is a non-integer real number and the second element is a natural number.

Analyzing the Right side of the Equation

The Set \((\mathbb{R} \times \mathbb{N})\) is the set of all ordered pairs, where the first element is a real number and the second element is a natural number. Now exclude \((\mathbb{Z} \times \mathbb{N})\) from it. This means all ordered pairs, in which the first element is an integer and the second is a natural number, are removed from \((\mathbb{R} \times \mathbb{N})\). Therefore, the remaining set contains all ordered pairs where the first element is a non-integer real number and the second element is a natural number.

Comparison

Both sides of the equation end up with the same set, thus the given statement is indeed true.

Key concepts

Set Theory

Set theory is a fundamental part of mathematics, dealing with the collection of objects, known as elements or members. It's essentially about understanding how different groups of items can be put together, compared, and related to one another. One of the most basic operations in set theory is the Cartesian product, which for two sets A and B, pairs every element of A with every element of B. The Cartesian product is denoted by the symbol \( \times \), and the resultant set consists of ordered pairs. Understanding set operations like union, intersection, and set subtraction is key to analyzing relationships between different sets.

In exercises involving set theory, it’s crucial to accurately interpret what sets represent and to maneuver between various set operations expertly. When it comes to a problem like \( (\mathbb{R}-\mathbb{Z}) \times \mathbb{N} = (\mathbb{R} \times \mathbb{N})-(\mathbb{Z} \times \mathbb{N}) \), recognizing that the subtraction of one set from another omits elements of the first set that are also in the second set, demonstrating a clear and methodical approach alleviates the complexity of arriving at the correct conclusion.

Real Numbers

The set of real numbers, denoted by \(\mathbb{R}\), includes all the numbers that can be found on the number line. This set encompasses rational numbers (which include integers \(\mathbb{Z}\) and fractions) and irrational numbers (those that cannot be expressed as a simple fraction, like \(\sqrt{2}\) or \(\pi\)). When discussing the Cartesian product in set theory, understanding the nature of these numbers is important, especially when differentiating them from integers. Real numbers that are not integers play a crucial role in creating unique ordered pairs during Cartesian product operations with sets like the natural numbers, which is explicitly used in the example problem provided. This distinction is at the heart of solving and verifying the truth of the statement given.

Natural Numbers

In the world of mathematics, natural numbers are potentially the most intuitive concept, represented by \(\mathbb{N}\), and include all positive integers beginning from 1 and moving upward (1, 2, 3, ...). These numbers are the counting numbers we use in everyday life. They exclude zero and all the negative numbers. In our Cartesian product example, \(\mathbb{N}\) serves as a crucial component because it pairs with real numbers to form ordered pairs. While artificial number concepts, like negative numbers or zero, are excluded from natural numbers, their absence significantly impacts the operations in set theory, particularly when deducing results of Cartesian products in problems such as the one mentioned.

Set Operations

Deep understanding of set operations is an essential aspect of working with sets. Think of sets as boxes containing certain items, and operations as ways to combine, modify, or compare these boxes. Key operations include:

• The Union (\(\cup\)), which gathers all unique elements from both sets.
• The Intersection (\(\cap\)), which finds common elements between sets.
• Set Subtraction (difference), noted by a minus sign (–), that excludes elements of one set from another.
• Cartesian Product (\(\times\)), combining elements from each set into ordered pairs, as highlighted in the exercise.

These operations are similar to arithmetic operations but applied to groups of objects instead of numbers. The exercise presented exemplifies how the sets formed by Cartesian products, and their subsequent subtraction, contribute to a deeper comprehension of how set operations interact. Symbols like \(\mathbb{R}\), \(\mathbb{Z}\), and \(\mathbb{N}\) are not just placeholders—they represent distinct sets of numbers that, when manipulated using set operations, reveal foundational relationships between them.