MATH 301 Homework 1 Solutions

MATH 301 Homework 1 Solutions - Math 301, Homework 1...

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Math 301, Homework 1 Solutions 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 ) : Solution 1 Suppose f is 1-2. Take two sets C 1 ; C 2 in S: Let y 2 f ( C 1 \ C 2 ) : Then there is some x 2 C 1 \ C 2 satisfying f ( x ) = y: But then x 2 C 1 and x 2 C 2 ; so y = f ( x ) 2 f ( C 1 ) and y = f ( x ) 2 f ( C 2 ) ; i.e. y 2 f ( C 1 ) \ f ( C 2 ) :: This shows that f ( C 1 \ C 2 ) ± f ( C 1 ) \ f ( C 2 ) : Conversely let y 2 f ( C 1 ) \ f ( C 2 ) : Then there are some x j 2 C j ; j = 1 ; 2 satisfying f ( x j ) = y: Since f is 1-1, we must have x 1 = x 2 : Then x 1 = x 2 2 C 1 \ C 2 ; so y = f ( x j ) 2 f ( C 1 \ C 2 ) : This shows that f ( C 1 ) \ f ( C 2 ) ± f ( C 1 \ C 2 ) : Then f ( C 1 \ C 2 ) = f ( C 1 ) \ f ( C 2 ) : Conversely suppose for all subsets C 1 ; C 2 in S we have f ( C 1 \ C 2 ) = f ( C 1 ) \ f ( C 2 ) : Suppose f ( x 1 ) = f ( x 2 ) = y for some x 1 6 = x 2 : Let C j = f
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MATH 301 Homework 1 Solutions - Math 301, Homework 1...

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