An integer set is a collection of numbers with no decimal or fractional part, including positive numbers, negative numbers, and zero. The set of all integers is often represented by the symbol Z.
Understanding Integers
An integer is a whole number. This means it can't have any digits after the decimal point. According to the provided reference, integers:
- Have no decimal or fractional part.
- Include negative numbers, positive numbers, and zero.
- Examples: -5, 0, 1, 5, 8, 97, and 3,043 are all integers.
Defining the Integer Set (Z)
The integer set, often denoted by Z, includes all possible integers, extending infinitely in both positive and negative directions. We can define Z more formally as:
Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
This set contains all whole numbers and their negatives. It's a fundamental concept in mathematics and is used extensively in various branches of the field.
Examples of Integer Sets
Here are a few examples to illustrate what constitutes an integer set:
- Example 1: A finite integer set: { -2, 0, 5, 10 } This set contains only integers.
- Example 2: An infinite integer set: { 1, 2, 3, 4, ... } This set contains all positive integers.
- Example 3: The set of solutions to the equation x2 - 4 = 0 where x is an integer. The solutions are x = 2 and x = -2. Thus, the integer set of solutions is { -2, 2 }.
What is Not an Integer Set?
A set is not an integer set if it contains any element that is not an integer. For example:
- { 1.5, 2, 3 } is not an integer set because 1.5 is not an integer.
- { 1, 2, 3.14159 } is not an integer set because 3.14159 (pi) is not an integer.
- { 1, 2, √2 } is not an integer set because √2 is not an integer.
Importance of Integer Sets
Integer sets are crucial in many areas of mathematics, including:
- Number Theory: The study of integers and their properties.
- Algebra: Integers form the basis for many algebraic structures.
- Computer Science: Integers are used extensively in programming and data structures.
