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home0sol

home0sol - x-z = x-y y-z Therefore if x-y is divisible by 2...

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Mth 261, Fall 2010 Homework Assignment 0 Solutions #2. The total space E is the set of all natural numbers from 1 to 10. A = { 1 , 2 , 3 , 4 } , B = { 2 , 4 , 6 , 8 } , C = { 3 , 4 , 5 , 6 } . 1. A B C = A ∩ { 4 , 6 } = { 4 } . 2. A - B = { 1 , 3 } . 3. ( A B ) c = { 1 , 2 , 3 , 6 , 8 } c = E - { 1 , 2 , 3 , 6 , 8 } = { 5 , 7 , 9 , 10 } . 4. A c B c C c = ( A B C ) c = { 4 } c = { 1 , 2 , 3 , 5 , 6 , 7 , 8 , 9 , 10 } . #5. A = { a, b } , B = { 2 , 3 } , C = { 3 , 4 } . 1. A × ( B C ) = A × { 2 , 3 , 4 } = { ( a, 2) , ( a, 3) , ( a 4) , ( b, 2) , ( b, 3) , ( b, 4) } . 2. ( A × B ) ( A × C ) = A × ( B C ) = { ( a, 2) , ( a, 3) , ( a, 4) , ( b, 2) , ( b, 3) , ( b, 4) } . 3. A × ( B C ) = { a, b } × { 3 } = { ( a, 3) , ( b, 3) } . 4. ( A × B ) ( A × C ) = A × ( B C ) = { ( a, 3) , ( b, 3) } . #9. This relation can be represented as the following set of ordered pairs: { (2 , 6) , (2 , 10) , (3 , 3) , (3 , 6) , (5 , 10) } . The domain of this relation is the set { 2 , 3 , 5 } while its range is the set F . Note that the image is the set { 3 , 6 , 10 } . #11. x y if ( x - y ) / 2, that is, x is in relation with y if x - y is divisible by 2. To show that this an equivalence relation we must show that: 1. x x for any natural number x . This is obviously true as x - x = 0 and it is divisible by 2. 2. If x y then y x . This is also true as it is true that if x - y is divisible by 2 so is y - x . 3. If x y and y z then

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Unformatted text preview: x-z = x-y + y-z . Therefore, if x-y is divisible by 2 and y-z is divisible by 2 then x-z = x-y + y-z is divisible by 2. 1 This proves that x ∼ y is an equivalence relation. #13. It should be easy to see that g ◦ f is such that ( g ◦ f )( a ) = t , ( g ◦ f )( b ) = s , ( g ◦ f )( c ) = t , ( g ◦ f )( d ) = r . #15. We have: f ( x ) = sin x , g ( x ) = e 2 x , h ( x ) = x 3 + 1. Therefore, ( f ◦ g )( x ) = sin( e 2 x ). ( g ◦ f )( x ) = e 2 sin x . ( f ◦ g ◦ h )( x ) = f ( g ◦ h )( x ) = f ( e 2( x 3 +1) ) = sin( e 2( x 3 +1) ). ( h ◦ g ◦ f )( x ) = h ( g ◦ f ) = h ( e 2 sin x ) = ( e 2 sin x ) 3 + 1 = e 6 sin x + 1. 2...
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home0sol - x-z = x-y y-z Therefore if x-y is divisible by 2...

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