Implication A statement of the form :

If p, then q.

is called an implication or a conditional statement.

Mathematicians have agreed that an implication will be called false only when the antecedent is true and the consequent is false. That is to say when P is true and q is false.

Function

It is important to note that this common notation is used only when .

Injective

A function is called injective (or one to one) if, for all implies that . An injective function is also referred to as an injection. (not necessary cover rng f).

Surjective

A function is called surjective (or one to one) if . A surjective function is also referred to as a surjection.

Bijective

A function is called bijective or bijection if it is both surjective and injective.

Inverse function

Let is bijective. The inverse function of is the function given by

Suppose that is any function. Then a function is called a

left inverse for if for all

right inverse for if for all

Cardinal numbers

Two sets S and T are called equinumerous, and we write S~T, if there a bijective function from S onto T.

A set S is said to be finite if or if there exists and a bijection . If a set is not finite, it is said to be infinite.

The cardinal number of is n, and if ~, we say that S has n elements. The cardinal number of is taken to be 0. If a cardinal number is not finite, it is called transfinite. (Cardinal number represents the size of a set.)

A set S is said to be denumerable if there exists a bijection If a set is finite or denumerable, it is called countable. If a set is not countable, it is uncountable. The cardinal number of a countable set is denoted by

Power set

Given any set S, let denote the collection of all the subsets of S. The set is called the power set of S.

Triangle inequality

Let , then

It is a direct observation of a triangle with the sum of length of two sides is greater than the remaining side, with inclusion of inequality for the case that when $x, y =0$.

Upper bound, lower bound, maximum and minimum

Let S be a subset of . If there exists a real number such that for all , then is called an upper bound for S. If for all , then is called an lower bound for S.

If an upper bound for S is a member of S, then is called a maximum of S, and we write .

If a lower bound for S is a member of S, then is called a minimum of S, and we write .

Supremum

If S is bounded above, then the least upper bound of S is called its supremum and is denoted by S . Thus S iff

(a)

and

(b) if , then there exists such that .

Completeness axiom

By this axiom, we made the fundamental difference between $latex\mathbb Q$ and . It states that

Every nonempty subset S of that is bounded above has a least upper bound. That is , S exists and is a real number.

Neighborhood

Let and let A neighborhood of is a set of the form

The number is referred to as the radius of

Deleted Neighborhood

Let and let A neighborhood of is a set of the form

Clearly,

Interior point and boundary point

Let . A point is an interior point of S if there exists a neighborhood of such that If for every neighborhood of , and , then is called a boundary point of S.

Closed Sets and Open Sets

Let . If then S is said to be closed. If then S is said to be open.

Cauchy sequence

A sequence is said to be a Cauchy sequence if for each there exists a number such that implies that .

Subsequence

Let be a sequence and let be any sequence of natural numbers such that The sequence is called a subsequence of .

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