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.  


It is important to note that this common notation f: A \to B is used only when dom f =A


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).


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.


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


 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,


(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. 


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


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<n_2<\cdots. The sequence {({s_n}_k)_{k=1}}^{\infty}is called a subsequence of {(s_n)_{n=1}}^{\infty}.