Homework 9 Solution - MATH 74 HOMEWORK 9 1. Note that if we...

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Unformatted text preview: MATH 74 HOMEWORK 9 1. Note that if we either remove or add exactly one element to a set with an odd number of elements, we get a set with an even number of elements. Thus the rule S 7 ( S- { 1 } 1 S S { 1 } 1 6 S defines a function f : X Y . It is also true that if we either remove or add exactly one element to a set with an even number of elements, we get a set with an odd number of elements. So the same rule just given also defines a function g : Y X . For any S X and T Y we have g ( f ( S )) = ( g ( S- { 1 } ) 1 S g ( S { 1 } 1 6 S = ( ( S- { 1 } ) { 1 } 1 S ( S { 1 } )- { 1 } 1 6 S = S and f ( g ( T )) = ( f ( T- { 1 } ) 1 T f ( T { 1 } 1 6 T = ( ( T- { 1 } ) { 1 } 1 T ( T { 1 } )- { 1 } 1 6 T = T we conclude that our function f : X Y is invertible, hence a bijection by a theorem from class. 2. Note that if Z is a 3-element subset of A , then A- Z = { a A : a 6 Z } is a 7-element subset of A...
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Homework 9 Solution - MATH 74 HOMEWORK 9 1. Note that if we...

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