MATH 301 Homework 1 - constructing such a function g; in...

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Math 301, Homework 1 1. Prove that for a function f : S ! T is one-to-one if and only if for all subsets C 1 ; C 2 in S we have f ( C 1 \ C 2 ) = f ( C 1 ) \ f ( C 2 ) : 2. Suppose that h : S ! T and k : T ! S are two functions such that h k = I T . Show that h is onto and k is one-to-one. Here I T is the identity function on T ; i.e. I T ( t ) = t for all t 2 T: Then the relation above reads as h ( k ( t )) = t for all t 2 T: 3. Show that (a) If f : S ! T is 1-1 then there is a function (left inverse) g : T ! S such that g f = I S . (b) If f : S ! T is onto then there is a function (right inverse) h : T ! S such that f h = I T . Hint: Showing that there is a function g ... means
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Unformatted text preview: constructing such a function g; in other words you should clearly say what g ( t ) is for a given t 2 T: The problem is trivial if f is a bijection. ‹t is a good idea to try to understand what is happenning by just writing a simple example where f is 1-1 or onto (but not both!) and work out to see how g; h could be de&ned. 4. Find a bijection f : Z ! N : (i.e. de&ne such a function and make sure that you check that f is 1-1 and onto.) 1...
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This note was uploaded on 06/25/2010 for the course MATH MATH 301 taught by Professor Alberterkip during the Fall '08 term at Sabancı University.

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