Beginning Rigorous Mathematics/Preliminaries

From Wikibooks, open books for an open world
Jump to: navigation, search

To study mathematics rigorously some key concepts must be understood. The most important of these is by far the groundwork which is laid down by basic logic. All rigorous mathematical arguments rely on this groundwork and it is therefore essential to understand and become familiar with basic logic. Second to this, is understanding mathematical objects (or just 'objects') - their definition and how they ultimately relate to other objects through logical statements. Thirdly it is important to understand the role and meaning of notation in writing rigorous mathematics unambiguously.

These three concepts seamlessly intertwine and any deficiency in one will certainly hamper the others.

Some initial definitions[edit]

Our initial definitions will mostly be intuitive rather than rigorous. More rigorous definitions form part of theories that are subjects of study on their own and the reader is encouraged to look at the formal definitions, but these are outside the scope of this text.

We will call a logical statement any (mathematical) statement that is either unambiguously true or unambiguously false. For example, "3 is less than 4" is a true logical statement, "9 is an even number" is a false logical statement, and "this statement is false" is not a logical statement.

Every statement has a negation. If a statement is true(false), its negation is false(true).

We define the word set to mean "a collection of distinct 'mathematical objects'" (the term 'mathematical objects' we leave undefined, but can be thought of any object that can be described mathematically). We say an object is an element of a set that contains it.

We define the word function to mean: "a rule of associating every element contained in a set, called the domain, unambiguously to exactly one element in another set, called the codomain"

Notation[edit]

The symbol ":=" is defined to read "equal by definition", and is used to define letters or symbols used to refer to commonly occurring objects. Statements involving the symbol ":=" are always assumed to be true. There is a subtle, but important difference between the symbols ":=" and "=". For example, we may first write "a:=4". This defines the symbol 'a' to equal 4, which is then assumed to be true. Then "a=5" and "a=4" are statements, the first of which is false and the second true.

Statements are often referred to by the letters P and Q.

Sets are often referred to by the capital letters A and B. If we want to show that a collection of objects form a set we will use curly braces to denote it. For example, \{1,2,3,4\} is the set containing 1,2,3 and 4.

We define the symbol "\in" to mean: "is an element of". So, x\in A reads "x is an element of (the set) A", and is a logical statement. So, for example 1\in\{1,2,3,4\} is a true statement and 5\in\{1,2,3,4\} is false.

Functions are often referred to by the letters f and g. If the set A is the domain, and the set B is the codomain of a function f we write f:A\rightarrow B which reads "f maps (the set) A into (the set) B". If x\in A we will use the notation f(x) to refer to the element in the codomain B which is associated to x through the function f.

Often a word in a statement is replaceable. When this is a case P is called a predicate. For example, we can define symbol P(x) to mean "x is an odd number". Then P(2) and P(5) are statements, the first of which is false and the second true.

We will only intuitively define the notion of a set as a collection of distinct mathematical objects called the elements of the set. Standard notation for writing sets are 'curly braces'. For example, we write A:=\{1,2,3,4\}, which means that 'A', by definition, refers to the set containing the elements '1', '2', '3' and '4'. We define the symbol "\in" to read "is an element of". Then "1\in A" and "5\in A" are statements, the first of which is true and the second false.

Sets can contain an infinite number of objects and can even contain sets themselves. Some of the most often used infinite sets are "the natural numbers" denoted by \mathbb{N}, "the rational numbers" denoted by \mathbb{Q}, and "the real numbers" denoted by \mathbb{R}.

Sets can also be defined be logical statements. The notation "\{x | P(x)\}" is defined to read "the set of all objects, x, such that P(x) is a true statement". For example, if as before, P(x):="x is an odd number", then \{x | P(x)\} is the set of all odd numbers. Then also, the statement "1234\in \{x | P(x)\}" is false and the statement "1233\in \{x | P(x)\}" is true.

Where A is any set, sometimes the notation "\{x\in A | P(x)\}" is also used and means "the set all elements, x, in set A such that P(x) is true".