Set Theory

A set is an unordered collection of distinct objects.

A distinct object refers to an item or entity that is clearly distinguishable from other items, meaning all elements within a set are unique.

Well-defined or distinct objects can include:

• numbers
• letters
• colors
• geometric shapes
• words

The notation \( \forall x \in P \), means for every x that is an element of set P.

The notation \( \forall x \in A, \text{ x is even} \), means for every x that is an element of set A, where x is even.

What are numbers?

A number is an arithmetic value used to represent quantity. So numbers are mathematical objects used to count, measure, and label the world.

There are different categories of numbers in mathematics, like Natural numbers \( \mathbb{N} \), Whole numbers \( \mathbb{W} \), and Integers \( \mathbb{Z} \).

This categorization of numbers evolved over time as our understanding of mathematics grew. So the concept of negative numbers and irrational numbers emerged as people encountered problems that could not be solved using only natural numbers or rational numbers.

Even though all of these sets fall under the umbrella of numbers, by categorizing them, we can study the properties and relationships unique to each set of numbers. Different number sets have distinct properties and characteristics, and categorizing them helps to specify which type of numbers we are working with in a given context. This allows for clear and precise communication when discussing mathematical concepts or solving problems.

Real numbers

Real numbers, denoted by the symbol \( \mathbb{R} \), encompass a broad set of numbers. Real numbers include all the numbers that can be represented on a number line, which means they can be positive, negative, zero, and also be rational or irrational.

\( \mathbb{R} : \{ -\infty, \infty \} \)

Natural numbers

Natural numbers or counting numbers, denoted by the symbol \( \mathbb{N} \), are a subset of whole numbers. Natural numbers include all positive integers (*whole numbers without any fractional or decimal parts*). They include the number 1, \( \mathbb{P} \)rime numbers, and composite numbers and go on infinitely in the positive direction. They are used for counting objects.

\( \mathbb{N} : \{ 1,2,3,... \} \)

Whole numbers

Whole numbers, denoted by the symbol \( \mathbb{W} \) or \( \mathbb{N}_0 \), are a subset of integers and include all the non-negative numbers without any fractional or decimal parts. They start from zero and go on infinitely in the positive direction.

\( \mathbb{W} : \{ 0,1,2,... \} \)

Integers

Integers, denoted by the symbol \( \mathbb{Z} \), are a subset of real numbers and include all whole numbers, both positive and negative, as well as zero. Integers can be represented on a number line and do not have any fractional or decimal parts. They are often used in mathematics to represent discrete quantities, such as the number of items in a set or the difference between two counts.

\( \mathbb{Z} : \{ ...,-3,-2,-1,0,1,2,3,... \} \)

The symbol \( \mathbb{Z} \) comes from the German word zahlen, which means numbers in English.

Rational numbers

A rational number, denoted by the symbol \( \mathbb{Q} \), is a number that can be written as a ratio of two integers. Basically, a number that can be expressed as the quotient or fraction of two integers with a non-zero divisor/denominator. Rational numbers include whole numbers, integers and fractions, like \( \frac{1}{2}, \frac{-3}{4}, \frac{6}{1} \).

If a number can be written as a fraction (the quotient of two integers with a non-zero denominator), it is rational, not irrational. Try simplifying the number to see if it can be expressed as a fraction.

In division operations:

• The top part is called the dividend.
• The bottom part is called the divisor.

In fractions:

• The top part is called the numerator.
• The bottom part is called the denominator.

Since any integer can be written as the ratio of two integers, all integers are rational numbers (*an easy way to write an integer as a ratio of integers is to write it as a fraction with denominator one*). Remember that the counting numbers and the whole numbers are also integers, and so they too, are rational.

In decimal form, a rational number will always terminate or repeat the same pattern of digits forever. For example, in decimal form: \( \frac{9}{11} \) is \( 0.818181... \), where the 81 repeats forever.

\( \mathbb{Q} : \{ \frac{p}{q} \mid p,q \in \mathbb{Z}, q \ne 0 \} \)

p over q such that p and q are a subset of \( \mathbb{Z} \) and q is not zero.

The symbol \( \mathbb{Q} \) comes from the word quotient, as rational numbers can be expressed as the quotient of two integers (with a non-zero denominator).

Irrational numbers

Irrational numbers, denoted by the symbol \( \mathbb{I} \) or \( \mathbb{R} \setminus \mathbb{Q} \), are numbers that cannot be expressed as a simple fraction. They have non-repeating, non-terminating decimal representations. Examples of irrational numbers include the \( \sqrt{2} \), the number PI \( \pi \) (approximately 3.14159265), the Golden Ratio \( \varphi \) (approximately 1.61803), and Euler’s number \( e \) (approximately 2.71828).

\( \mathbb{I} : \{ x \mid x \in \mathbb{R}, \text{ x is not rational } \} \)

Irrational numbers and rational numbers are two distinct categories of real numbers, and they are mutually exclusive. A number cannot be both irrational and rational at the same time.

When we see an irrational number in decimal form, the number will not terminate nor repeat the same pattern forever, so it cannot be written as a fraction.

Although \( \pi \) is defined as the ratio of a circle's circumference to its diameter, it is still an irrational number, as it cannot be precisely represented as a simple fraction with integer values in the numerator and denominator.

Something to be aware of when using a calculator for like \( \sqrt{5} \), the decimal representations are often truncated or rounded for practical purposes, but the true value continues infinitely without repeating.

\( \sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}, \sqrt{7}, \sqrt{8} \) are all non-terminating, non-repeating decimals which cannot be written as the quotient of two integers. You can also multiply irrational numbers to generate more of them...

\( \mathbb{R} \setminus \mathbb{Q} \) means the set of real numbers excluding the set of rational numbers, where the backward slash symbol denotes 'set minus'.

Complex Numbers

Complex numbers, denoted by \( \mathbb{C} \), extend the concept of real numbers by incorporating an additional component, the imaginary unit. A complex number is expressed in the form \( z = a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit defined by \( \sqrt{-1} = i \).

\( \mathbb{C} : \{ a + bi \mid a, b \in \mathbb{R} \} \)

Summary

As we work our way up the chain of rational numbers, we can see that a number such as 5, could be classified as a rational number (since \( \frac{5}{1} = 5 \)), an integer, a whole number, and a natural number.

So all natural numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers.

But reversing this isn't always going to work.

-5 is a rational number and an integer, but it's not a whole number nor a natural number.

\( \frac{3}{4} \) is a rational number, but not an integer, whole number, nor natural number.