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Unformatted text preview: 1.246 1. (a) Clearly ( x ) ( x ) and ( y ) ( y ). Consequently ( x ) ( y ) ( x ) ( y ). We claim that these sets are in fact equal. Let z ( x ) ( y ). Suppose that z ( x ) but z / ( x ). Then z x . By transitivity, z x y which implies that z ( y ). Similarly z ( y ) ( y ) implies z ( x ). Therefore ( x ) ( y ) = ( x ) ( y ) (b) By continuity, ( x ) ( y ) is open and ( x ) ( y ) = ( x ) ( y ) is closed. Further x y implies that x ( y ) so that ( x ) ( y ) = . We have established that ( x ) ( y ) is a nonempty subset of which is both open and closed. Since is connected, this implies (Exercise 1.83) that ( x ) ( y ) = 2. (a) By definition, x / ( x ). So ( x ) ( y ) = implies x ( y ), that is x y contradicting the noncomparability of...
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- Fall '10