# Adv Alegbra HW Solutions 38 - is the inclusion then there...

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38 1.102 My Uncle Ben was born in Pogrebishte, a village near Kiev, and he claimed that his birthday was February 29, 1900. I told him that this could not be, for 1900 was not a leap year. Why was I wrong? Solution. Even though 1900 was not a leap year in America, it was a leap year in Russia, which did not adopt the Gregorian calendar until after the Russian Revolution. Exercises for Chapter 2 2.1 True or false with reasons. (i) If S T and T X , then S X . Solution. True. (ii) Any two functions f : X Y and g : Y Z have a composite f g : X Z . Solution. False. (iii) Any two functions f : X Y and g : Y Z have a composite g f : X Z . Solution. True. (iv) For every set X ,wehave X × = . Solution. True. (v) If f : X Y and j : im f Y
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Unformatted text preview: is the inclusion, then there is a surjection g : X → im f with f = j ◦ g . Solution. True. (vi) If f : X → Y is a function for which there is a function g : Y → X with f ◦ g = 1 Y , then f is a bijection. Solution. False. (vii) The formula f ( a b ) = ( a + b )( a − b ) is a well-de f ned function Q → Z . Solution. False. (viii) If f : N → N is given by f ( n ) = n + 1 and g : N → N is given by g ( n ) = n 2 , then the composite g ◦ f is n 7→ n 2 ( n + 1 ) . Solution. False. (ix) Complex conjugation z = a + ib 7→ z = a − ib is a bijection C → C . Solution. True....
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