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Mathematics - Algebraic Number Theory - Pigeonhole Principle

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1a. If X and Y are infinite sets with the same number of elements, show that the following conditions are equivelent for a function f: X-->Y:

(i). f is injective
(ii). f is bijective
(iii) f is surjective

1b. Suppose there are 11 pigeons sitting in some pigeonhole. If there are only 10 pigeonholes prove that there is a hole containing more than one pigeon.

I just dont know where to begin with the proof and how to show this.

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Solution Summary

The expert examines algebraic number theory for the Pigeonhole Principle. A Complete, Neat and Step-by-step Solution is provided in the attached file.

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Let us prove this by contradiction.
Let P be the set of pigeons and H be the set of holes.
Assume that every hole has at most one pigeon in it.
We define f: P  ...

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