North, South, East,Westnorth south east west.The four cardinal directions are, north, east, south , and west.
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
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.
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.
Googleplex to the tent powerr!! NO DUR!!!
North, South, East,Westnorth south east west.The four cardinal directions are, north, east, south , and west.
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
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.
To draw an E-R diagram for school fee management, identify the main entities involved such as the students, fees, payments, and classes. Establish the relationships between these entities by adding appropriate cardinalities and connect them with lines. Add attributes to each entity, such as student ID, fee amount, payment date, etc. Additionally, include any additional entities and relationships, like invoice generation or fee waivers, that are specific to the school's fee management process.
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.
It is based on collection of objects.An object contain values stored in instance variables within the object.An object also contains the bodies of code that operate on object.These bodies of code is called methods.Collection of instance variables and method is called classes.
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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