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.
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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.