The set E={…,-4,-2,0,2,4, …} of even integers can be...

The set $E=\{\ldots,-4,-2,0,2,4, \ldots\}$ of even integers can be described by means of a defining condition by $E=\{y=2 x: x \in \mathbf{Z}\}=\{2 x: x \in \mathbf{Z}\}$. Describe the following sets in a similar manner. (a) $A=\{\ldots,-4,-1,2,5,8, \ldots\}$ (b) $B=\{\ldots,-10,-5,0,5,10, \ldots\}$ (c) $C=\{1,8,27,64,125, \ldots\}$

The set $E=\{\ldots,-4,-2,0,2,4, \ldots\}$ of even integers can be described by means of a defining condition by $E=\{y=2 x: x \in \mathbf{Z}\}=\{2 x: x \in \mathbf{Z}\}$. Describe the following sets in a similar manner.
(a) $A=\{\ldots,-4,-1,2,5,8, \ldots\}$
(b) $B=\{\ldots,-10,-5,0,5,10, \ldots\}$
(c) $C=\{1,8,27,64,125, \ldots\}$

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Key Concepts

Set Builder Notation

This is a method of defining a set by specifying a property that its members must satisfy. Instead of listing every element, one provides a rule (often in the form of an equation or inequality) that generates exactly the desired elements using a variable that ranges over a specified domain, such as the set of integers.

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. Such sequences can be expressed in the form {ax + b : x ? Z} where a is the common difference and b is the starting offset, allowing for a concise description of sets with evenly spaced elements.

Polynomial Mappings

Polynomial mappings are functions constructed from variables using only addition, multiplication, and non-negative integer exponents. For example, the cubic function, defined as x^3, maps natural numbers (or integers) to their cubes. This concept is useful in describing sets whose elements are generated by raising numbers to a particular power.

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