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Q: What is Cardinalities?

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Googleplex to the tent powerr!! NO DUR!!!

It's the number of mappings, *or* he number of available objects to map something to, *or*...See also http://en.wikipedia.org/wiki/Cardinality

North, South, East,Westnorth south east west.The four cardinal directions are, north, east, south , and west.

It means without limit, a sequence that goes on and on forever.But if you really want to get into it, there are different "levels" of infinity: or infinities with different cardinalities.

Assuming that you mean within the context of a relational database, the process would typically go something like this: 1) Identify in general terms what the database is supposed to do - for example, a hotel reservation system. 2) Determine the sort of things the system needs to keep track of - in our example, this would include rooms, guests, reservations, stays, etc. You also would identify the relations between these things - a guest makes a reservation for a room and so on. A frequent tool to do this is called an Entity-Relationship or ER Diagram. 3) Design the general structure of the database with respect to the ER diagram - each entity usually will become one or more database tables and the relationships frequently denote foreign key relations between tables. Also identify cardinalities (one-to-one, one-to-many, many-to-many, etc.) Note that many-to-many relationships will require additional tables to normalize. 4) Flesh out the tables with the various data elements in each and specify data constraints (primary and foreign keys, data domains, etc.) At this point, you may have enough information to begin creating the actual database.

In math, infinity is not a unique concept.There are different ways of looking at infinity.One is to consider that something is true when a limit is taken of larger and larger numbers.Example: if x>0 then 1/x > 0, but if x>N it follows that 1/x < 1/N and so we can make 1/x as small as we want. This is written as lim(1/x, x=infinity) = 0.Example: if you throw dice, then the average is defined as sum(x_i, i = 1..n)/n. A limit theorem says that the expected value of the dice, defined by lim(sum(x_i, i = 1..n)/n, n=infinity) exists and equals sum(1/6 * i, i = 1..6) = 3.5.--Another concept of infinity arizes when you start counting the elements of a set.The number of elements of a set A is called its cardinality and is written as card(A) or |A|. The "number of elements" formulation works with finite sets, but not with infinite sets.Therefore, more precisely, the cardinalities of two sets A and B are considered equal if and only if there exists a bijection f:A->B.f:A->B is a bijection between A and B if:- for all b in B there is an a in A such that f(a) = b.- for all a1, a2 in A with a1a2, f(a1)f(a2).This definition of cardinality defines an equivalence relation between sets and it causes sets to be classified as belonging to a class of sets with equal cardinality.There is also a natural order in the cardinalities.card(A)

The easiest way might to answer that might be to show you an example. Let's look at a chess board with two of the diagonally opposite corners removed. Is it possible to cover the board with pieces of domino whose size is exactly two board squares? The reason this is a pigeonhole problem is because the two diagonal square on a chess board are the same color. So when you remove them you have 2 more square of one color than you do of the other. So assume by contradiction that you can cover the board with pieces of domino whose size is exactly two board squares. Now every piece of domino must cover exactly two squares and these will be squares of different colors because adjacent square on the chess board are different colors. So for every domino piece I place, I set up a 1 to 1 correspondence between the set of one color square and the set of the other color squares. We now know the cardinality of the two sets is different since we removed those corners. So the pigeonhole principle tells us we can not have a 1 to 1 correspondence between two sets with different cardinalities. We conclude that it can't be done. The idea in all cases where you want to use the pigeonhole principle and prove by contradiction is to assume it works and then let the pigeonhole principle prove it can't work.

Infinity is not a specific number but a cardinality. The cardinality of a finite set is the number of elements in the set. For example, the cardinality of the set {1, 2, 3, 4, 5} is 5. Simple enough, but what about the cardinality of all natural numbers? There is no end to natural numbers so the cardinality cannot be a number in the normal sense. The cardinality is an infinity, called aleph-null. [As an aside, aleph is the first letter of the Hebrew language – which, along with the next letter, beth, gives us the word alphabet.]The cardinality of any set which can be put into one-to-one correspondence with the set of natural numbers (or conversely) is also aleph-null. You then have the curious result that, using the mappings x-> 2x-1 and x -> 2x the cardinality of positive odd number is also aleph-null as is the cardinality of positive even numbers. Comparing cardinalities, you get the aleph-null + aleph-null = aleph-null or 2* aleph-null = aleph-null.This result can be extended to all integers so that n * aleph-null = aleph-null for all integers n. This leads to the counter-intuitive result that aleph-null * aleph-null = aleph-null! It is possible to devise a diagonal scheme which gives a one-to-one correspondence between all rational numbers and all natural numbers. So there are aleph-null rational numbers. The classic exposition for this is Hilbert’s Grand Hotel. See https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_HotelThe above sets are said to have countably infinite elements (since they can be put into 1-1 correspondence with the counting numbers.You may have noted that, when introducing aleph-null, I used the phrase “an infinity”. This is because there is another, higher cardinality: the uncountably infinite. The cardinality of the set of all subsets of a set with countably infinite elements, or 2-to-the-power-aleph-null. This infinity is also known as the continuum. Cantor proved that the cardinality of irrational numbers (and therefore the real numbers) is the continuum and also that there are no orders of infinity between aleph-null and the continuum.

Object-based Logical ModelsObject-based logical models:Describe data at the conceptual and view levels.Provide fairly flexible structuring capabilities.Allow one to specify data constraints explicitly.Over 30 such models, includingEntity-relationship model.Object-oriented model.Binary model.Semantic data model.Infological model.Functional data model.At this point, we'll take a closer look at the entity-relationship (E-R) and object-orientedmodels.The E-R ModelThe entity-relationship model is based on a perception of the world as consisting of a collection of basic objects(entities) and relationships among these objects.An entity is a distinguishable object that exists.Each entity has associated with it a set of attributesdescribing it.E.g. number and balance for an account entity.A relationship is an association among several entities.e.g. A cust_acct relationship associates a customer with each account he or she has.The set of all entities or relationships of the same type is called the entity set or relationship set.Another essential element of the E-R diagram is the mapping cardinalities, which express the number of entities to which another entity can be associated via a relationship set.We'll see later how well this model works to describe real world situations.The overall logical structure of a database can be expressed graphically by an E-R diagram:rectangles: represent entity sets.ellipses: represent attributes.diamonds: represent relationships among entity sets.lines: link attributes to entity sets and entity sets to relationships.See figure 1.2 for an example.Figure 1.2: A sample E-R diagram.The Object-Oriented ModelThe object-oriented model is based on a collection of objects, like the E-R model.An object contains values stored in instance variableswithin the object.Unlike the record-oriented models, these values are themselves objects.Thus objects contain objects to an arbitrarily deep level of nesting.An object also contains bodies of code that operate on the the object.These bodies of code are called methods.Objects that contain the same types of values and the same methods are grouped into classes.A class may be viewed as a type definition for objects.Analogy: the programming language concept of an abstract data type.The only way in which one object can access the data of another object is by invoking the method of that other object.This is called sending a message to the object.Internal parts of the object, the instance variables and method code, are not visible externally.Result is two levels of data abstraction.For example, consider an object representing a bank account.The object contains instance variables number and balance.The object contains a method pay-interest which adds interest to the balance.Under most data models, changing the interest rate entails changing code in application programs.In the object-oriented model, this only entails a change within the pay-interest method.Two objects containing the same values are distinct.Distinction is created and maintained in physical level by assigning distinct object identifiers.Unlike entities in the E-R model, each object has its own unique identity, independent of the values it contains:

Infinity is not just really big number - and consequently the concepts of rational vs irrational cannot be applied to it. It is a marvelously useful concept with great utility in mathematics but don't confuse it for being the same as a number that we could write out and categorize just because we have a symbol for representing it. When you stick infinity into an equation you get things like "limits" rather than a fixed answer; for example - for the function f(x) = (x-1)/x, if x = ∞ you don't actually get a value for the function- rather you get a limit that it approaches as x goes off to infinity; in this case the limit as x approaches infinity is 1. For the function f(x) = (x-2)/x, the limit as x approaches infinity is ALSO 1, and for the function f(x) = x/(x-1) the limit as x approaches infinity is .... 1. Obviously for any finite number they will not have the same value, but conceptually they all converge to the same value as you go to infinity. Hopefully this illustrates why you cannot apply the concept of rational vs irrational to "infinity".

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