YES
here is the proof: http://planetmath.org/encyclopedia/ProductOfAFiniteNumberOfCountableSetsIsCountable.html
CHECK THIS OUT http://www.mathstat.dal.ca/~hill/2112/assn7sol.pdf
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.
The cardinality of finite sets are the number of elements included in them however, union of infinite sets can be different as it includes the matching of two different sets one by one and finding a solution by matching the same amount of elements in those sets.
The union of two sets.The union of two sets.The union of two sets.The union of two sets.
here is the proof: http://planetmath.org/encyclopedia/ProductOfAFiniteNumberOfCountableSetsIsCountable.html
Yes, finite numbers are always countable.
CHECK THIS OUT http://www.mathstat.dal.ca/~hill/2112/assn7sol.pdf
all finite set is countable.but,countable can be finite or infinite
A product measure is sigma-finite if each of its component measures is sigma-finite. This means that for each component measure, the space can be decomposed into a countable union of measurable sets, each with finite measure. Consequently, when taking the product of these measures, the resulting product measure retains this property, allowing for the entire space to be covered by countably many sets of finite measure. This is crucial for the application of Fubini's theorem in integrating functions over product spaces.
It is a measure, but it isn't always sigma-finite. Take your space X = [0,1], and u = counting measure if u(E) < infinity, then E is a finite set, but there is no way to cover the uncountable set [0,1] by a countable collection of finite sets.
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.
Proof By Contradiction:Claim: R\Q = Set of irrationals is countable.Then R = Q union (R\Q)Since Q is countable, and R\Q is countable (by claim), R is countable because the union of countable sets is countable.But this is a contradiction since R is uncountable (Cantor's Diagonal Argument).Thus, R\Q is uncountable.
The cardinality of finite sets are the number of elements included in them however, union of infinite sets can be different as it includes the matching of two different sets one by one and finding a solution by matching the same amount of elements in those sets.
They are sets with a finite number of elements. For example the days of the week, or the 12 months of the year. Modular arithmetic is based on finite sets.
Closed sets and open sets, or finite and infinite sets.
the number of steps of an algorithm will be countable and finite.