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Can the set of mixed numbers is the union of the set of integers and the set of fraction?
No, the set of mixed numbers is a subset of the set of rational numbers. For example the mixed number 1 ¼ is the same as the improper fraction 5/4 [a rational number]. Note that it is a subset, because integers are also rational numbers, but a mixed number will not be an integer. Also, any fraction between 0 and 1 will not be a mixed number.
The union of the counting numbers and zero are called what?
The set of whole numbers.
Is the set of all irrational number countable?
No, the set of all irrational numbers is not countable. Countable sets are those that can be put into a one-to-one correspondence with the natural numbers (1, 2, 3, ...). The set of irrational numbers is uncountable because it has a higher cardinality than the set of natural numbers. This was proven by Georg Cantor using his diagonalization argument.
What is the set of whole numbers and their opposites called?
Assuming that you mean opposites (rather than oppisites), the answer will depend on what kind of opposites: additive opposites or multiplicative opposites.In the first case you have the same set as you started with: whole numbers.In the second, you will have the set that consists of the union ofall whole numbers,all unit fractions between -1 and +1. But, you have a problem with zero: its multiplicative opposite is not defined.
Are the natural numbers a Borel set?
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.
