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Q: Is it possible for a finite set to have the same cardinality as its power set?

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A Natural infinite set refers to one whose members can be put into 1-to-1 correspondence with the natural numbers, while a real infinite set is one whose members can be put into 1-to-1 correspondence with the real numbers. Although both sets are infinite, they are not of the same cardinality (size).The cardinality of the natural infinite set is denoted by Ã€0 or Aleph-null. The cardinality of the real infinite set is 2 to the power Ã€0, which is denoted by C. (Actually Aleph looks like an N with wriggly lines but this browser is incapable of displaying it.)For more on the cardinality of infinite sets, see the related links. Georg Cantor's diagonal argument is exquisite - simple but immensely powerful. If you want to get a feel for transfinite arithmetic - read about Hilbert's Hotel paradox.

They are not. A line segment is of finite length whereas a ray is infinitely long.

Two sets are equivalent if they have the same cardinality. For finite sets it simply means that the two sets have the same number of distinct elements. Thus {1, 2, 3, 3} is equivalent to {a, b, b, b, b, c, c}; each set has cardinality 3. For infinite sets, it is a bit more complicated. If you can define a bijective mapping (one-to-one and onto) from the elements of one set to the elements of the other then the two sets are equivalent. By definition, bijective mappings must have an inverse so defining the mapping in one direction is sufficient. It is relatively easy to see that the set of odd integers is equivalent to the set of even integers if you consider the mapping f(x) = x+1 where x is odd. Or perhaps f(x) = x + 7 What is less obvious is that the set of all integers is equivalent to the set of even integers when you consider the mapping f(x) = 2x where x is any integer. Thus it is possible for a proper subset of an infinite set to be equivalent to the superset!

This is NOT true.The cardinality of the set of points in a circle is the same as the cardinality of the set of points in a line.First, break the circle and straighten it out. I think you would agree that the number of points remains the same.Now apply some continuous monotonic function that takes one end of that line segment and assigns it to -infinity and the other end to +infinity. I think you would agree that this is possible.We have now made a one-to-one, invertible correspondence between the points in the original circle and the points in a line, demonstrating that the two objects have the same cardinality.Roughly speaking!

Hopefully the same thing it means in most other grades of geometry, which is a straight one dimensional object that is infinitely long. It differs from a ray, which has a finite point and extends infinitely from that point in a single direction, and a line segment, which has two finite ends.

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Two sets are equivalent if they have the same cardinality. For finite sets this means that they must have the same number of distinct elements. For infinite sets, equal cardinality means that there must be a one-to-one mapping from one set to the other. This can lead to some counter-intuitive results. For example, the cardinality of the set of integers is the same as the cardinality of the set of even integers although the second set is a proper subset of the first. The relevant mapping is x -> 2x.

The cardinality of [0,1) is equal to the cardinality of (0,1) which has the same cardinality as the real numbers.

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.

In SQL (Structured Query Language), the term cardinality refers to the uniqueness of data values contained in a particular column (attribute) of a database table.The lower the cardinality, the more duplicated elements in a column. Thus, a column with the lowest possible cardinality would have the same value for every row. SQL databases use cardinality to help determine the optimal query plan for a given query.

Actually, infinity is not a natural number. It is simply a concept of having no upper bound. However, it is possible to have and compare different infinities. For example, we use aleph_0 to represent the cardinality (size) of the set of natural numbers. The cardinality of the set of integers, rational numbers, gaussian integers all have the same cardinality of aleph_0. The set of real numbers has cardinality aleph_1, which is greater than aleph_0. It is possible to create a sequence of increasing infinities (aleph_2, aleph_3, ...), which are called transfinite numbers.

Equivalent sets are sets that have the same cardinality. For finite sets it means that they have the same number of distinct elements.For infinite sets, though, things get a bit complicated. Then it is possible for a set to be equivalent to a proper subset of itself: for example, the set of all integers is equivalent to the set of all even integers. What is required is a one-to-one mapping, f(x) = 2x, from the first set to the second.

No. The "smallest" infinity is the cardinality of the natural numbers, N. This cardinality is named Aleph-null. Rational numbers also have the same cardinality as do n-tuples of rational numbers. The next larger cardinality is that of the real numbers. This is the "continuum, C, which equals 2aleph-null. As with the cardinality of the natural numbers, n-tuples of reals have the same cardinality. The point about introducing n-tuples, is that they are used to denote points in n-dimensional space. If you want more read some of the Wikipedia articles of Cantor, Hilbert's Grand Hotel. These could lead you to many more related articles - though sadly, not infinitely many!

Infinity squared is infinity. But there's more to it.Mathematicians describe different kinds of infinities:The cardinality(number) of natural numbers is called Aleph0 () . This is infinite, and it has some peculiar properties:The cardinality of even numbers is also Aleph0.As is the cardinality of odd numbers.As is the cardinality of rational numbers (which you could view as infinity squared, but it still gives you infinity.The cardinality of countable ordinal numbers is called Aleph1 ().The cardinality of the real numbers is two to the exponent of Aleph0 ( ). The continuum hypothesis says this is equal to Aleph1.Basically, if you square an infinite set from a given cardinality, the cardinality stays the same (meaning Aleph0 squared is still Aleph0, etc.)If your mind just burst(cause mine did! 0_o), do not worry. This is a common reaction to set theory.See the related link for more on Aleph numbers, which are how mathematicians view infinity.

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.

yes, it is possible. in fact in power systems all the generators do not run with same power factor.

Curiously, both sets are countably infinite and so their cardinality is the same.

There is no one to one correspondence between the real numbers and the set of integers. In fact, the cardinality of the real numbers is the same as the cardinality of the power set of the set of integers, that is, the set of all subsets of the set of integers.

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