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a cardinal is a type of bird that is read with a black looking mask around its eyes
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What are mapping cardinalities?
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
What are the 4 cardinalities?
North, South, East,Westnorth south east west.The four cardinal directions are, north, east, south , and west.
What does infinitely mean in math terms?
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.
What is infinity and its implications?
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)
How can you solve pigeonhole questions using contradiction?
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.
