devotion & mathematics

 PROOFS OF FEW RESULTS  1:                    Here we are going to see the proofs of some results what we know. They are   1.  Intersection of an infinite collection of open sets is not open.  2. The set N of natural numbers is not open.  3.  Every neighborhood is an open set.  4.  Union of any collection of open sets is open. Now come to proofs : 1. INTERSECTION OF AN INFINITE COLLECTION OF OPEN SETS IS NOT OPEN : PROOF : 2. THE SET N OF NATURAL NUMBERS IS NOT OPEN: PROOF :  3. EVERY NEIGHBORHOOD IS AN OPEN SET : PROOF : 4. UNION OF ANY COLLECTION OF OPEN SETS IS OPEN : PROOF : people also ask         * what is an interior point * what is an open set        * what is a neighborhood                * what is extended real number                      *...

contents : 1. Extended real number system 2. The symbol infinity 3. Properties of infinity THE EXTENDED REAL NUMBER SYSTEM :                                    Basically the real number system not contain + or -  ∞, it  contains all the real numbers lies between for - ∞ and + ∞ i.e.  the real number system R=(- ∞. ∞).                                     The extended real number system is a union of the real number set R and the symbols - ∞ and + ∞ i.e. Extended real system = R ⋃ {- ∞, ∞} = [- ∞, ∞]    In extended real number system,  ∞ is an upper bound   and - ∞ is a lower bound of every subset. Here we see an example to tell the main difference between real and extended real number systems.     Consider an open interval (2, ∞).  The interval (2, ∞) i...

Real  Number System :                              In mathematics mainly there are two kinds of number systems. They are  1. Real Number System. 2. Complex Number System.    About complex number system we discuss later.  Here I introduce about what is a Real number system i.e. what are called the real numbers. The real number system includes basically  the set of all rational and irrational numbers i.e the set of real numbers is an union of set of rational numbers and the set of irrational numbers. But there are some subsets are there for the set of all rational numbers. They are set of natural numbers, set of whole numbers, set of integers.  Therefore the set of Real Numbers is a combination of  1. set of natural numbers 2. set of whole numbers                                    ...