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A finite set is one that contains a specific, limited number of elements, while a countable set can be either finite or infinite but can be put into a one-to-one correspondence with the natural numbers. In other words, a countable set has the same size as some subset of the natural numbers, meaning it can be enumerated. For example, the set of all integers is countable, even though it is infinite, whereas the set of all even integers is also countable.
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What are finite 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.
What is the kinds of sets?
Closed sets and open sets, or finite and infinite sets.
Give you examples of finite sets of numbers?
sets
What is countable set and uncountable set?
A countable set is defined as one whose elements can be put into one-to-one correspondence with elements of the set of counting numbers or some subset of it. A countable set can be infinite: for example all even numbers. This raises the strange concept where a subset (positive even numbers) has the same cardinality as all counting numbers - which should be a set that is twice as large! Even more confusingly (perhaps) is the fact that the set of all rational numbers also has the same cardinality as the set of counting numbers. You need to go to the set of irrationals or bigger before you get to uncountable sets. So you have the weird situation in which there are more irrationals between 0 and 1 than there are rationals between from 0 and infinity (if infinity can be treated as a value)! There is a minority definition of countable which means containing a finite number of elements as opposed to uncountable meaning infinitely many elements. However, these definitions are essentially the same as the finite sets and infinite sets and so there is little point in using them.
What is the two kind of sets?
Closed sets and open sets, or finite and infinite sets.
Is the union of finite countable sets finite?
YES
Prove that a finite cartesian product of countable sets is countable?
here is the proof: http://planetmath.org/encyclopedia/ProductOfAFiniteNumberOfCountableSetsIsCountable.html
Is counting measure indeed a measure and is this always sigma-finite?
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.
What does it mean by countable plate?
A countable plate refers to a type of mathematical object in set theory, where a set is considered countable if its elements can be put into a one-to-one correspondence with the natural numbers. This means that even if the set is infinite, it can still be "counted" in the sense that its elements can be listed sequentially. Countable sets include finite sets and countably infinite sets, such as the set of integers or rational numbers. In some contexts, "countable plate" might also refer to a specific type of surface or geometric object, but the term is less commonly used in that sense.
How product measure is sigma finite?
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.
What are finite 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.
What is the kinds of sets?
Closed sets and open sets, or finite and infinite sets.
Show that the union of two countable sets is countable?
CHECK THIS OUT http://www.mathstat.dal.ca/~hill/2112/assn7sol.pdf
Give you examples of finite sets of numbers?
sets
What is countable set and uncountable set?
A countable set is defined as one whose elements can be put into one-to-one correspondence with elements of the set of counting numbers or some subset of it. A countable set can be infinite: for example all even numbers. This raises the strange concept where a subset (positive even numbers) has the same cardinality as all counting numbers - which should be a set that is twice as large! Even more confusingly (perhaps) is the fact that the set of all rational numbers also has the same cardinality as the set of counting numbers. You need to go to the set of irrationals or bigger before you get to uncountable sets. So you have the weird situation in which there are more irrationals between 0 and 1 than there are rationals between from 0 and infinity (if infinity can be treated as a value)! There is a minority definition of countable which means containing a finite number of elements as opposed to uncountable meaning infinitely many elements. However, these definitions are essentially the same as the finite sets and infinite sets and so there is little point in using them.
What is the two kind of sets?
Closed sets and open sets, or finite and infinite sets.
What is kind of set?
Closed sets and open sets, or finite and infinite sets.
