Question

Write each of the following sets in set-builder notation. $$ \left\\{\ldots, \frac{1}{27}, \frac{1}{9}, \frac{1}{3}, 1,3,9,27, \ldots\right\\} $$

Short answer

The set can be written in set-builder notation as \(\{3^n : n \in Z\}\)

Step-by-step solution

Identify the Pattern

Firstly, observe the numbers in the set which consist of 1, 3 and its powers. These numbers include fractions which are actually inverse powers of 3, such as \(1/27 = 3^{-3}\), \(1/9 = 3^{-2}\), \(1/3 = 3^{-1}\). 1 is actually \(3^0\), 3 is \(3^1\), 9 is \(3^2\) and so on. The pattern repeats indefinitely in both directions.

Write in set-builder notation

Set-builder notation is represented as \{ expression : condition \}, where the expression is a rule which demonstrates how the numbers in the set are generated, and the condition limits the numbers included in the set. From the pattern observed, we can see that the expression should be \(3^n\), and the condition is that n is an integer, because the power to which 3 is raised can be negative, zero or positive integer.

Key concepts

Mathematical Sets

Sets are a fundamental concept in mathematics, representing a collection of distinct objects, often referred to as elements or members. The beauty of sets lies in their ability to group objects into a well-defined collection, which can be finite or infinite, and they can contain numbers, characters, or even other sets. When describing sets, mathematicians often use set-builder notation to express the properties that all members of a set share succinctly.

For example, the set in our exercise includes numbers that are powers of 3 or inverse powers of 3. In set-builder notation, which is a concise way to define sets using a rule or condition, we describe our set based on the property that each element 'x' can be written as a power of 3. This illustrative method is efficient for expressing complex or infinite sets, such as the one given in the exercise. We use variables and conditions in braces to clearly define the set's criteria without listing every single element.

Exponents and Powers

The concept of exponents and powers is pivotal in understanding various mathematical and scientific phenomena. An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in the expression \(3^2\), the base is 3, and the exponent is 2, indicating that 3 is to be multiplied by itself once (since the first instance of 3 is implied), resulting in 9.

Furthermore, exponents can also be negative or zero. A negative exponent indicates an inverse power, which means that the base is on the bottom of a fraction instead of the top. Zero as an exponent is special because any nonzero base raised to the power of zero equals one. These rules are crucial for understanding the pattern of numbers in our exercise because the set includes both positive and negative powers of 3 as well as the zero power.

Mathematical Notations

Mathematical notations are the shorthand of math, allowing us to convey information efficiently and unambiguously. Notations are symbols or combinations of symbols that represent mathematical objects, operations, relations, and properties. In our exercise, we encounter several important mathematical notations.

Set-builder notation is illustrated by the use of curly braces and a colon, which separates the general expression of the elements from the condition they must satisfy. Within this framework, the exponent notation is also employed to concisely represent multiplication of the base number by itself. Being proficient in interpreting and using these notations is key to understanding and solving mathematical problems. They are the language in which mathematical ideas are expressed, making them as crucial to mathematics as words are to poetry.