Question

Write each of the following sets in set-builder notation. $$ \\{0,1,4,9,16,25,36, \ldots\\} $$

Short answer

The set can be written in set-builder notation as: \( \{x \in \N : x = n^2\} \)

Step-by-step solution

Identify the Properties of the Set Elements

First, observe the given set: \( \{0,1,4,9,16,25,36, \ldots\} \). Notice that the elements of the set are perfect squares of natural numbers, starting from 0.

Describe the Set in Set-Builder Notation

Set-builder notation describes a set by specifying the properties that its members must satisfy. This set can be described as 'the set of all numbers that are the square of a natural number'. In symbolic form, this becomes: \( \{x \in \N : x = n^2\} \) , where \( \N \) represents the set of natural numbers. This reads as 'the set of all \( x \) in \( \N \) such that \( x = n^2 \) '.

Key concepts

Perfect Squares

Perfect squares are the product of a natural number multiplied by itself. For example, 1, 4, 9, and 16 are perfect squares because they can be expressed as \( 1^2 \), \( 2^2 \), \( 3^2 \), and \( 4^2 \) respectively. The concept of perfect squares is fundamental in various areas of mathematics, including algebra and number theory. Perfect squares have several interesting properties:

• Every perfect square has an odd number of positive factors. This is because the factors come in pairs, except for the square root, which is counted once.
• The difference between consecutive perfect squares is always an odd number. For instance, \( 4 - 1 = 3 \) and \( 9 - 4 = 5 \).
• A number is a perfect square if and only if its prime factorization has even powers of all primes.

These properties are helpful for identifying perfect squares and understanding their characteristics within a set.

Natural Numbers

Natural numbers are the set of positive integers starting from 1. In mathematical symbols, this set is often represented by the capital letter \( \textbf{N} \) or \( \textbf{N}^+ \). It includes numbers like 1, 2, 3, and so on. These numbers are used for counting and ordering. There are a few important points to keep in mind about natural numbers:

• Natural numbers do not include zero or negative numbers.
• They also exclude fractions and decimals – only whole numbers belong to this set.
• In some mathematical contexts, the set of natural numbers is considered to include zero. It's always good to double-check how the term natural numbers is being used.

Natural numbers are the building blocks of many other sets of numbers, like integers, rational numbers, and real numbers.

Mathematical Notation

Mathematical notation is a system of symbols used to write out equations and formulas across various fields of mathematics. It provides a concise and precise way to communicate mathematical ideas. For instance:\
\

\
• \( + \), \( - \), \( \times \), and \( \textbf{div} \) represent the basic arithmetic operations.
\
• The symbol \( \textbf{N} \) often denotes the set of natural numbers, while \( \textbf{Z} \) represents integers.
\
• Equalities and inequalities are expressed using symbols like \( = \), \( e \), \( < \), and \( \textbf{‘x’} \).
\

\
Understanding mathematical notation is crucial for working with formulas, equations, and proving theorems. This language of symbols allows mathematicians to describe complex ideas in a standardized format that is universally understood within the community.

Sets in Mathematics

Sets are a fundamental concept in mathematics. A set is a collection of distinct objects, considered as an object in its own right. Sets can contain numbers, characters, or even other sets. Here are some key points:\
\

\
• A set is defined by its elements, which can be anything from numbers to letters to other sets.
\
• Two sets are considered equal if they contain exactly the same elements.
\
• An element either belongs to a set or does not, there is no in-between.
\

\
In set-builder notation, we describe a set by specifying a property that characterizes its elements. For example, the set of perfect squares can be written in set-builder notation as \( \textbf{‘S’} \).