Question

Write out the indicated sets by listing their elements between braces. $$ \\{n \in \mathbb{Z}: 2

Short answer

The Cartesian product of the two sets is \{(3, -5), (3, 5), (4, -5), (4, 5)\}.

Step-by-step solution

Identify elements in the first set

For the first set, the condition is \(n\) in \(\mathbb{Z}\): 2<\(n\)<5. So, you need to list all integers \(n\) such that \(n\) is greater than 2 and less than 5. The only integers that satisfy this condition are 3 and 4. Therefore, the first set is \{3, 4\}.

Identify elements in the second set

For the second set, the condition is \(n\) in \(\mathbb{Z}\): |\(n\)| = 5. So, you need to list all integers \(n\) such that the absolute value of \(n\) is 5. The integers that satisfy this condition are -5 and 5. Therefore, the second set is \{-5, 5\}.

Find the Cartesian product of the two sets

The Cartesian product of two sets A and B is a set of ordered pairs (a, b) where a belongs to A and b belongs to B. Therefore, the Cartesian product of \{3, 4\} and \{-5, 5\} is \{(3, -5), (3, 5), (4, -5), (4, 5)\}.

Key concepts

Set Theory

When we talk about set theory, we delve into the fundamental language of mathematics. It deals with the concept of a 'set', which is simply a collection of distinct objects, considered as an object in its own right. These objects are called 'elements' or 'members' of the set. Sets are typically denoted by curly braces, with elements listed between them, such as \( \{1, 2, 3\} \).

Let's look at an example related to our exercise. Given a set description such as \( \{n \in \mathbb{Z}: 2
In set theory, two important operations are 'union' and 'intersection'. Union (denoted by \(\cup\)) combines all elements of the involved sets, whereas intersection (denoted by \(\cap\)) finds elements common to all sets involved. Neither of these are to be confused with the Cartesian product, which is about pairing elements rather than combining them within a single set.

Absolute Value

The absolute value of a number is a measure of its magnitude, regardless of its sign. In mathematical terms, the absolute value function is denoted by two vertical lines as \( |x| \). It is defined as \(x\) itself if \(x\) is greater than or equal to zero and \( -x \) if \(x\) is less than zero.

For instance, in our textbook exercise, we look for all integers \(n\) such that \( |n| = 5 \). This means we want the positive and negative integer whose magnitude is 5. The result is two numbers, -5 and 5, because both have an absolute value of 5. When teaching absolute value, it is critical to emphasize that it represents distance from zero, and thus, is always non-negative.

Understanding the concept of absolute value is vital in various branches of mathematics, from solving equations to analyzing real-life situations, such as finding the difference in elevation regardless of whether you're moving up or down from sea level.

Integers

Integers are the set of whole numbers that include positive numbers, negative numbers, and zero. Notationally, we refer to the set of all integers as \(\mathbb{Z}\), which encompasses \(\{..., -3, -2, -1, 0, 1, 2, 3, ...\}\).

In the context of our problem, we focus on integers within specific ranges or with particular properties (like having an absolute value of 5). This inclusion and exclusion of numbers are what define our sets. It's important for students to differentiate integers from other number types, such as rational numbers (which can be expressed as fractions) and real numbers (which include rational numbers as well as irrational numbers).

Moreover, when working with sets of integers, keep in mind the arithmetic rules and properties unique to them, like the division of integers, where the remainder becomes a critical aspect of the operation.