Confusing concepts in Sets & Sequences

Hello reader! There are various basic concepts in Mathematics such as-Difference between Open & closed sets, Relation between sets & sequences,Partial sum,which must be clear before learning mathematics, otherwise the subject seems disaster to understand.. In this post we are going to discuss about such basic concepts..

We begin with the most basic concepts-Difference between Open & closed sets:

Difference between Open & Closed Sets:

The notation of Open set is-(a,b), & the closed set is-[a,b]. Now, we see the difference between Open & Closed Sets:

  • The meaning of this (a,b) notation is that the elements in this set are between a & b. i.e. a< m< b, (m is any element in Open set.)
  • The meaning of this [a,b] notation is that the elements in this set are between a & b also, it includes a & b. i.e. a= m= b, (m is any element in Open set.)
  • There is also a concept-“Semi open Set” or “Semi closed set” which is any of these-(a,b]- a left open interval, [a,b)- a right open interval.
  • A set is Open, if it contains its Limit Points.
  •  if the set does not contain its Limit Points, then it is Close Set.

About Sequences:

There are chain of relations behind the set & sequences which is quite complicate to remember, made easy here..

  • If the limit of the sequence, say {X} exist then it is-
    1. Continuous
    2. Convergent
    3. Compact  (If the sequence has convergent sub sequence converging in the set itself.)
    4. Complete(If the seq. is Cauchy seq.)
    5. Bounded
  • Continuity implies boundedness but boundedness do not imply continuity.
  • But if the mapping is identity, them the boundedness imply continuity.
  • A sequence is absolutely convergent if the set along with its mod is convergent, i.e. if the sequence {Xn} is convergent & its mod i.e.|{Xn}| is also convergent, then {Xn} is absolutely convergent. Otherwise, {Xn} is conditionally convergent.

Basic concepts (Remember):

  • Partial sum is nothing but a part of the total summation.
  • When mod (|.|) separates then the then the less than or equal to(=)sign replaces the equal to sign(=).
  • When norm (||.||) separates then the less than or equal to(=)sign replaces the equal to sign(=).
  • If the Sequence {s} is bounded, then its range(set containing terms of corresponding sequence) is also bounded.
  • The sum of two divergent sequences need not to be divergent.
  • The product of two divergent sequences need not to be divergent.
  • If <sn>, <tn>, <un> be three sequences, such that- <sn>= <tn>= <un> ; & lim<sn> = lim<un> = l, then- ?lim<sn>=l.
  • If the sequence <sn> converges to l, then its each coset converges to l.
  • If a monotone increasing (decreasing) sequence bounded above, then it converges to the supremum(infimum).

This is all about the concepts in sequences, further discussion will be in next post.. Hope it’ll be useful to you..