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 $f: A \to B$ is used only when $dom f =A$

Injective

A function  $f: A \to B$ is called injective (or one to one) if, for all $a, a' \in A, f(a)=f(a')$ implies that $a=a'$. An injective function is also referred to as an injection. (not necessary cover rng f).

Surjective

A function  $f: A \to B$ is called surjective (or one to one) if  $B=rng f$. A surjective function is also referred to as a surjection.

Bijective

A function  $f: A \to B$ is called bijective or bijection if it is both surjective and injective.

Inverse function

Let  $f: A \to B$ is bijective. The inverse function of $f$ is the function $f^{-1}$ given by

$f^{-1}=\{(y,x)\in B*A:(x,y) \in f\}.$

Suppose that $f: A \to B$ is any function. Then a function $g: B \to A$ is called a

left inverse for $f$ if $g(f(x))=x$ for all $x\in A$

right inverse for $f$ if $f(g(y))=y$ for all $y\in B$

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 $S=\varnothing$ or if there exists $n \in \mathbb N$ and a bijection $f: \{1,2,\cdots,n\}\to S$. If a set is not finite, it is said to be infinite.

The cardinal number of $I_n$ is n, and if $S$~$I_n$, we say that S has n elements. The cardinal number of $\varnothing$ 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 $f:\mathbb N\to S.$ 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 $\aleph_0.$

Power set

Given any set S, let $\mathcal P(S)$ denote the collection of all the subsets of S. The set $\mathcal P(S)$ is called the power set of S.

Triangle inequality

Let $x, y \in \mathbb R$, then

$|x+y| \leq |x|+|y|$

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 $\mathbb R$. If there exists a real number $m$ such that $m \geq s$ for all $s\in S$, then $m$ is called an upper bound for S. If $m \leq s$ for all $s\in S$, then $m$ is called an lower bound for S.

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

If a lower bound for S is a member of S, then $m$ is called a minimum of S, and we write $m=min S$

Supremum

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

(a) $m\geq s, \forall s\in S,$

and

(b) if $m' < m$, then there exists $s'\in S$ such that $s'>m'$

Completeness axiom

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

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

Neighborhood

Let $x\in \mathbb R$ and let $\epsilon>0.$ A neighborhood of $x$ is a set of the form

$N(x;\epsilon)=\{y\in\mathbb R: |x-y|<\epsilon \}.$ The number $\epsilon$ is referred to as the radius of $N(x,\epsilon).$

Deleted Neighborhood

Let $x\in \mathbb R$ and let $\epsilon>0.$ A neighborhood of $x$ is a set of the form

$N*(x;\epsilon)=\{y\in\mathbb R: 0<|x-y|<\epsilon \}.$

Clearly,  $N*(x;\epsilon)=N(x;\epsilon)\ \{x\}.$

Interior point and boundary point

Let $x\in \mathbb R$. A point $x\in \mathbb R$ is an interior point of S if there exists a neighborhood $N$ of $x$ such that $N \subseteq S.$ If for every neighborhood $N$ of $x$, $N \bigcap S \neq \varnothing$ and $N \bigcap (\mathbb R/S) \neq \varnothing$, then $x$ is called a boundary point of S.

Closed Sets and Open Sets

Let $S\subseteq \mathbb R$. If $bd S\subseteq S,$then S is said to be closed. If $bd S\subseteq \mathbb R/S,$then S is said to be open.

Cauchy sequence

A sequence $(s_n)$ is said to be a Cauchy sequence if for each $\varepsilon>0$ there exists a number $N$ such that $m, n>N$ implies that $|s_n -s_m|<\varepsilon$

Subsequence

Let  ${(s_n)_{n=1}}^{\infty}$ be a sequence and let ${(n_k)_{k=1}}^{\infty}$be any sequence of natural numbers such that $n_1 The sequence ${({s_n}_k)_{k=1}}^{\infty}$is called a subsequence of ${(s_n)_{n=1}}^{\infty}$.

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