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What definition proves that the set of all odd natural numbers is a countable set?
It is NOT a 'countable set'. It is an infinite set. 1, 3, 5, 7, 9, 11, ... you can count to infinity and keep going.
How to determine the domain set of all real numbers?
By definition, it is the set of all real numbers!
The set of all rational and irrational numbers?
Are disjoint and complementary subsets of the set of real numbers.
What is the Difference between a finite set and countable set?
all finite set is countable.but,countable can be finite or infinite
What is a natural number that is not real?
The set of Natural Numbers is the set of 'counting numbers' {1,2,3,4,....}. All of them are also real numbers.
Is the set of all irrational number countable?
No, it is uncountable. The set of real numbers is uncountable and the set of rational numbers is countable, since the set of real numbers is simply the union of both, it follows that the set of irrational numbers must also be uncountable. (The union of two countable sets is countable.)
Is set R of real numbers is countable set or not?
It is uncountable, because it contains infinite amount of numbers
Can set of rational numbers forms a borel set?
Yes. the set of rational numbers is a countable set which can be generated from repeatedly taking countable union, countable intersection and countable complement, etc. Therefore, it is a Borel Set.
Are the real numbers a borel set?
Yes, since the set of real numbers can be expressed as a countable union of closed sets.In fact if we're talking about subsets of the real numbers (R), then by definition R is in all sigma-algebras of R including the Borel sigma-algebra, and so is a Borel set.
What do you mean by countably infinite and infinite?
Countably infinite means you can set up a one-to-one correspondence between the set in question and the set of natural numbers. It can be shown that no such relationship can be established between the set of real numbers and the natural numbers, thus the set of real numbers is not "countable", but it is infinite.
What is the set of numbers that includes all rational and all irrational numbers?
the set of real numbers
What definition proves that the set of all odd natural numbers is a countable set?
It is NOT a 'countable set'. It is an infinite set. 1, 3, 5, 7, 9, 11, ... you can count to infinity and keep going.
What is the set of numbers including all irrational and rational numbers?
real numbers
How to determine the domain set of all real numbers?
By definition, it is the set of all real numbers!
Derived Set of a set of Rational Numbers?
The derived set of a set of rational numbers is the set of all limit points of the original set. In other words, it includes all real numbers that can be approached arbitrarily closely by elements of the set. Since the rational numbers are dense in the real numbers, the derived set of a set of rational numbers is the set of all real numbers.
What is the set of all real numbers?
The set of all real numbers (R) is the set of all rational and irrational numbers. The set R has no restrictions in its domain and so includes (-∞, ∞).
Do numbers go on forever?
The question is a bit vague. The set of all natural numbers N (0,1,2,...) has no 'end', there is no 'largest number', in other words: it has an infinite amount of elements. The set of all real numbers R (which includes -2,sqrt(3), pi, e, 56/8, etc.) als has infinitely many elements, but there is a difference between the two: N is a countable set (you can 'count' all the elements), but R is not. If you want to know more about this, you should search after terms like cardinality, countable set, aleph, ...
