Real Number Properties

Real numbers have certain behaviors when exposed to different operations. In Mathematics, sometimes we are dealing with subsets of Real Numbers and we define these subsets in different terms such as Groupoid, Semi-Group, Monoid, Group and others. These will be discussed in detail in Chapter 2. However, these classifications are defined by which properties can be applied on them.

Some of these propeties are closure, associative, commutative, distributive, identity, and inverse properties.

Operations

All mathematical properties are defined in terms of some operation. There are one, two, or more inputs and usually there is one output. Addition, subtraction, multiplication, division, union, intersection, complement, logarithm, exponentiation, negation, factorial, reciprocal, and absolute are all operations you might have heard about or used or will use in this course.

The properties we will be looking at here will be for binary operations.

Binary Operation

Addition, subtraction, multiplication, division, union, intersection are all binary operations. Where two numbers interact with each other to produce a third number. Binary operations will be properly defined in chapter 2. They can be much more complex than simple addition or multiplication. For now, we will only be discussing the properties in the context of addition and multiplication as simple binary operations.

Closure Property

If the closure property is applied to a set for a specific operation, we say that "This set is closed w.r.t. (with respect to) this operation". Spoiler, The Real Number set is closed w.r.t. to both addition, multiplication and subtraction.

What does being closed mean?

It means that applying an operation yields a number from the same set. If we add 2 and 3, both natural numbers, is the resulting number also a natural number? Yes. If this is true for all possible natural numbers such that you add any two natural numbrs and the result is always a natural number, then the natural number set is closed w.r.t. addition.

What does not being closed mean?

It means that applying an operation on members of a set yield a new number outside of the set. If we subtract 3 from 2, we get -1. -1 is not a natural number. So we say that the natural number set is not closed w.r.t. subtraction.

Some other examples

These statements are true, try and figure out why.

• The set of rational numbers is closed w.r.t. multiplication
• The set of irrational numbers is not closed w.r.t. multiplication
• The set of natural numbers is not closed w.r.t. reciprocal
• The set of natural numbers is not closed w.r.t. negation
• The set of rational numbers is not closed w.r.t. reciprocal

How do we denote closure property mathematically?

Addition

This reads as, For all a and b in the Real Number set, a added to b will also be in the Real Number set.

Addition

This reads as, For all a and b in the Real Number set, a multiplied by b will also be in the Real Number set.

General Binary Operation

This reads as, For all a and b in the Real Number set, a operated on by b (or a asterisk b) will also be in the Real Number set.

How do you say a set is not closed?

Lack of Closure for Subtraction in Natural Numbers

This reads as, There exists some a and b in the Natural Number set, such that a minus b does not belong in the Natural Number set.