Mathematical Proof and the Principles of Mathematics/Preliminaries
The study of mathematics is founded on logic. All rigorous mathematical arguments rely on logic and it is therefore essential to understand and become familiar with basic logic.
Built on this foundation are mathematical concepts. The nature of these concepts is described by logical statements, consisting of a set of fundamental statements called axioms, and the statements that logically follow from them. Some mathematical concepts are defined in terms of more basic concepts, but there is a bottom level where the concepts are simply undefined. At that point, only the relationships between the concepts, as given by the axioms, are given. This means that the ultimate nature of mathematical objects is unknowable, and there can be many different interpretations.
In the study of mathematics it is also important to understand the role and meaning of notation and terminology, the language used to write about mathematics. As much as possible, this language is precise and unambiguous, which makes it different than natural languages as the are normally used. Because of this difference, it takes some time and practice to get used to mathematical language, much as with learning a second language.
Some initial definitionsEdit
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"
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 and .
Sets are often referred to by the capital letters and . If we want to show that a collection of objects form a set we will use curly braces to denote it. For example, is the set containing 1,2,3 and 4.
We define the symbol " " to mean: "is an element of". So, reads " is an element of (the set) ", and is a logical statement. So, for example is a true statement and is false.
Functions are often referred to by the letters and . If the set is the domain, and the set is the codomain of a function we write which reads " maps (the set) into (the set) ". If we will use the notation to refer to the element in the codomain which is associated to through the function .
Often a word in a statement is replaceable. When this is a case is called a predicate. For example, we can define symbol to mean "x is an odd number". Then and 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 , which means that 'A', by definition, refers to the set containing the elements '1', '2', '3' and '4'. We define the symbol " " to read "is an element of". Then " " and " " 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 , "the rational numbers" denoted by , and "the real numbers" denoted by .
Sets can also be defined be logical statements. The notation " " is defined to read "the set of all objects, x, such that P(x) is a true statement". For example, if as before, "x is an odd number", then is the set of all odd numbers. Then also, the statement " " is false and the statement " " is true.
Where is any set, sometimes the notation " " is also used and means "the set all elements, x, in set A such that P(x) is true".