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.
An example is given here: http://en.wikipedia.org/wiki/Non-Borel_set Any set that is easy to think of will be a Borel set, so an example of a non-Borel set will be complicated. Another approach: All Borel sets are Lebesgue measurable. The axiom of choice can be used to give an example of a non-measurable set, and this set will also be a non-Borel set. See http://en.wikipedia.org/wiki/Non-measurable_set = =
The set of Rational Numbers is a [proper] subset of Real Numbers.
Real numbers
What are equal sets?? A set is a grouping of numbers. Set P = {1,4,9} if set Q is equal it must contain exactly the same numbers.
A singleton point is a closed set. The natural numbers can be written as a countable union of points. Thus, they form 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.
One example of a simple Borel measurable function is the indicator function of a Borel set. This function takes the value 1 on the set and 0 outside the set, making it easy to determine its measurability with respect to the Borel sigma algebra.
Pierre-Arnold Borel was a French mathematician known for his work in set theory and real analysis. He is best known for his contributions to measure theory and the Borel algebra.
real numbers
An example is given here: http://en.wikipedia.org/wiki/Non-Borel_set Any set that is easy to think of will be a Borel set, so an example of a non-Borel set will be complicated. Another approach: All Borel sets are Lebesgue measurable. The axiom of choice can be used to give an example of a non-measurable set, and this set will also be a non-Borel set. See http://en.wikipedia.org/wiki/Non-measurable_set = =
In a certain sense, the set of complex numbers is "larger" than the set of real numbers, since the set of real numbers is a proper subset of it.
the set of real numbers
Are disjoint and complementary subsets of the set of real numbers.
Real numbers are a proper subset of Complex numbers.
The set of real numbers is a subset of the set of complex numbers. For the set of complex numbers, given in the form (a + bi), where a and b can be any real number, the number is only a real number, if b = 0.
There is only one set of Real numbers.