Question

Write each of the following sets in set-builder notation. $$ \\{3,4,5,6,7,8\\} $$

Short answer

The set \( \{3,4,5,6,7,8\} \) in set-builder notation is \( \{x | 3 \leq x \leq 8, x \in \mathbb{Z}\} \)

Step-by-step solution

Identify Structure

Look at the provided set and identify any patterns. It's clear that the set encompasses a series of consecutive integers, starting from 3 and ending at 8.

Identify Common Property

Identifying a property that all these numbers share, it can be seen that all numbers are integers that fall within a certain range. This can be denoted by stating a number \( x \) must be greater than or equal to 3 and less than or equal to 8.

Write in set-builder notation

Using the set-builder notation, this common property can be written as such: \( \{x | 3 \leq x \leq 8, x \in \mathbb{Z}\} \). This represents the set of all values \( x \) such that \( x \) is an integer and \( x \) is greater than or equal to 3 and less than or equal to 8.

Key concepts

Mathematical Sets

A mathematical set is essentially a collection of distinct objects, considered as an object in itself. These objects, called elements or members, can be anything: numbers, people, letters, etc. Sets are particularly useful in mathematics because they provide a way to group and organize information.

For example, the set from the exercise is \( \{3, 4, 5, 6, 7, 8\} \), which is a collection of numbers. When describing sets, it's important to be precise about the characteristics that define what is in and what is out of the set. That's where set-builder notation comes in handy, providing a clear and concise way to represent sets based on a specific property or rule.

Consecutive Integers

Consecutive integers are numbers in a sequence, each number being one more than the previous. They follow a consistent pattern without any gaps between them. Recognizing consecutive integers is essential in many areas of mathematics, including algebra and number theory.

In the context of the textbook exercise above, the numbers in the set \( \{3, 4, 5, 6, 7, 8\} \) are consecutive integers starting at 3 and ending at 8. Whenever you see a series of numbers increasing by one, you can communicate that as a range from the smallest to the largest, just as it was identified in step 1 of the solution.

Integer Properties

Integers are the set of whole numbers and their opposites including zero (\( \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \)). They can be positive, negative, or zero but do not include fractions or decimals. Integers have certain properties that distinguish them from other numbers. For instance, they are evenly spaced on the number line, and every number within them is a unit distance apart from the next.

The property that all numbers in the given set are integers is an important one—it's what allows us to use the symbol \( \mathbb{Z} \) in set-builder notation to specify that we are only interested in those whole number values within a certain range.

Set Notation

Set notation is a standardized mathematical language used to describe sets. There are several types of set notations, but one of the most compact and informative ways to convey the information about a set is set-builder notation. This notation encapsulates the idea of a set with an intrinsic property or a rule that all its elements obey.

In the original exercise, the set-builder notation for the set \( \{3, 4, 5, 6, 7, 8\} \) is given by \( \{x | 3 \leq x \leq 8, x \in \mathbb{Z}\} \), where \( x \) represents the elements of the set. This reads as: 'the set of all \( x \) such that \( x \) is an integer and \( x \) ranges from 3 to 8, inclusive.' It elegantly combines all aspects touched upon in earlier sections, like the nature of the numbers, their consecutive property, and the fact that they are integers.