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The simple form of it states:
If m pigeons are put into m pigeonholes, there is an empty hole iff there's a hole with more than one pigeon.
In more formal math language it says:
Let |A| denote the number of elements in a finite set A ( also known as its cardinality). For two finite sets A and B, there exists a 1-1 correspondence f: A -->B if and only if |A| = |B|.
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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.
Use pigeonhole principle to show that one of any n consecutive integers divisible by n?
Let's look at a simple example first such as 3 consecutive integers. We want to show that it is divisible by 3.Take 4,5, and 6 and of course 6 is divisible by 3.The reason for this is can be seen using the pigeonhole principle.When an integer is divided by 3, possible remainders are 0, 1, and 2. It follows that everyinteger can be expressed in one of the forms 3k, 3k + 1, and 3k + 2 where k is an integer.So if we have any three consecutive integers, one of them must be divisible by 3.Let's look at how the pigeonhole principle applies. Suppose we have 3 consecutive integers are non are divisible by 3. Think of the pigeon holes as 3k, 3k + 1, and 3k + 2, now this means no numbers are in the 3k hole and two of them must be in either the 3k+1 hole or the 3k+2 hole. But this contradicts that they are consecutive integers.So for any n, let the pigeon holes be nk, nk+1,... nk+(n-1) and these exhaust the multiples of n. Now if you take n consecutive numbers, you must have a least 1 number in the nk pigeon hole or else they are not consecutive.
What is a mathematical principle?
Counting Principle is one of them
What mathematical principle is equal to 3.14?
That's not a "mathematical principle", it is an approximation of the number pi.That's not a "mathematical principle", it is an approximation of the number pi.That's not a "mathematical principle", it is an approximation of the number pi.That's not a "mathematical principle", it is an approximation of the number pi.
What is the zero principle of multiplication?
the principle is 24*0=0
