COUNTABLE, ATMOST COUNTABLE & PERFECT SETS #countable #atmost #perfect #sets #uncountable
COUNTABLE SETS:-
Consider N, the set of all +ve integers. Any set A is called countable of ∃ a bijection between A and N.
Examples:-
1. The set Z of all integers is countable.
2. The set N of all +ve integers is countable.
3. The set Q of all rational numbers is countable.
4. The set of elements {(1/n) / nεN} is countable.
5. The set of all even +ve integers is countable.
6. The set of all odd +ve integers is countable.
Properties of countable sets:-
1. Any subset of a countable set is countable.
2. Any subset of N is countable.
3. Union of a finite number of countable sets is countable.
4. A union of countable sets is countable.
Atmost countable sets:
A set which is either finite or countably infinite is called atmost countable.
properties:-
1. every finite set is atmost countable.
2. An atmost countable set is nearly countable.
Uncountable sets:-
A set which is not countable is uncountable.
Examples:-
1. The real line R is uncountable.
2. The set R-Q, of all irrational numbers is uncountable.
3. The cantor set is uncountable.
4. Every interval [a, b] is uncountable.
5. A set of all sequences whose elements are 0 & 1 is uncountable.
Perfect sets:-
A non-empty set A is called a perfect set if A is closed and every point of A is a limit point of A.
i.e., if A=D(A) then A is called a perfect set.
Examples:-
1. The real line R is a perfect set.
2. All closed intervals [a, b] are perfect.
3. The cantor set is perfect.
4. The empty set ⲫ is also considered as a perfect set.
Counter examples:-
1. Finite sets are not perfect.
2. Sets with isolated points are not perfect.
3. Sets with missing limit points are not perfect.
Properties of perfect set:-
1. Every non-empty perfect set is uncountable.
2. A perfect set cannot have isolated points.
people also ask
1. What is an interior of a set?
2. What are called interior points?
3. What ae called limit points?
4. What are axioms of real number system?
5. What is extended real number system?
6. What is convergence of a sequence?
#countable #atmost #perfect #sets #uncountable
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