Homework 9 Solution

# 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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