By definition, the two sets do not overlap. This is because the irrationals are defined as the set of real numbers that are not members of the rationals.
Negative integers, integers, negative rationals, rationals, negative reals, reals, complex numbers are some sets with specific names. There are lots more test without specific names to which -10 belongs.
Yes, every Cauchy sequence of real numbers is convergent. In other words, the real numbers contain all real limits and are therefore continuous, and yes the integers are discrete in that the set of integers only contains (very very few, with respect to the set of rationals) rational numbers, i.e. their values can always be accurately displayed unlike the set of reals which is dense with irrational numbers. It's so dense with irrationals in fact, that by comparison, the set of rationals can be called a null set, however that is a different topic.
All the elements in a group must be invertible with respect to the operation. The element 0, which belongs to the set does not have an inverse wrt multiplication.
There are rational numbers and irrational numbers. Real numbers are DEFINED as the union of the set of all rational numbers and the set of all irrational numbers. Consequently, all rationals, by definition, must be real numbers.
It could be the set denoted by Q- (the non-positive rationals) or Q+ (the non-negative rationals).
Integers, rationals, reals, complex numbers, etc.
The number -4 belongs to the set of all integers. It also belongs to the rationals, reals, complex numbers.
By definition, the two sets do not overlap. This is because the irrationals are defined as the set of real numbers that are not members of the rationals.
There is not a specific abbreviation. The set is denoted by R - Q: the real numbers minus the rationals.
The answer depends on what do you mean by "all". It could be the set of all integers, the set of all rationals or the set of all reals.
There are an infinity of possible answers: the integers, rationals, reals, complex numbers, the set {0,1,-3}, the set containing only the element 0;
Q represents the set of all rational numbers, Zrepresents the set of all integers so Q excluding Z, represents all rationals that are not integers.
There are an infinite number of subsets: All rationals other than 1 All rationals other than 2, etc All rationals other than 1.1 All rationals other than 2.1, etc, etc. All integers
The Rationals, the set {1, 3 , 5.86, sqrt(59), -2/3, pi2}, the reals numbers, numbers between 5 and 6, etc.
Negative integers, integers, negative rationals, rationals, negative reals, reals, complex numbers are some sets with specific names. There are lots more test without specific names to which -10 belongs.
There are more irrational numbers than rational numbers. The rationals are countably infinite; the irrationals are uncountably infinite. Uncountably infinite means that the set of irrational numbers has a cardinality known as the "cardinality of the continuum," which is strictly greater than the cardinality of the set of natural numbers which is countably infinite. The set of rational numbers has the same cardinality as the set of natural numbers, so there are more irrationals than rationals.