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Unformatted text preview: M114S and M134, solutions to HW #6 x6.2. Suppose that S is a chain in the function space ( A → E ). We need to find a least upper bound for S . For each a ∈ A , let S ( a ) = { f ( a )  f ∈ S } . Then we show that for each a ∈ A , S ( a ) is a chain in E . Let f ( a ) and g ( a ) both belong to S ( a ). Since S is a chain, either f ≤ g or g ≤ f in the pointwise order. Either way, we have f ( a ) ≤ g ( a ) or g ( a ) ≤ f ( a ). For each a ∈ A , let a ∗ = sup S ( a ). Let h : A → E be given by h ( a ) = a ∗ . We claim that h = sup S . For all f ∈ S , f ≤ h because for each a ∈ A , f ( a ) ∈ S ( a ) and hence f ( a ) ≤ a ∗ . This shows that h is an upper bound of S in the function space. Let g be any upper bound of S , we show h ≤ g . To do this, fix a ∈ A . Then as g is an upper bound of S , g ( a ) ≥ f ( a ) for all f ∈ S . So g ( a ) ≥ a ∗ , as a ∗ = sup S ( a ). So g ( a ) ≥ h ( a ). This for all a shows that h = sup S . x6.3. Suppose C ⊆ P 1 × P 2 is a chain, and let C 1 = { x 1 ∈ P 1  ( ∃ x 2 ∈ P 2 )[( x 1 ,x 2 ) ∈ C ] , C 2 = { x 2 ∈ P 2  ( ∃ x 1 ∈ P 1 )[( x 1 ,x 2 ) ∈ C ] . Claim: C 1 is a chain in P 1 . Proof of Claim: If x 1 and x ′ 1 are in C 1 , then there exist x 2 and x ′ 2 such that ( x 1 ,x 2 ) ∈ C and ( x ′ 1 ,x ′ 2 ) ∈ C ; since C is a chain, these two pairs must be comparable, i.e., either ( x 1 ,x 2 ) ≤ ( x ′ 1 ,x ′ 2 ) or ( x ′ 1 ,x ′ 2 ) ≤ ( x 1 ,x 2 ) (with the partial ordering on P 1 × P 2 ). Now in the first case we have x 1 ≤ 1 x ′ 1 , and in the second case we have x ′ 1 ≤ 1 x 1 . This completes the proof of the Claim....
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This homework help was uploaded on 04/16/2008 for the course MATH 114S taught by Professor Moschovakis during the Winter '08 term at UCLA.
 Winter '08
 MOSCHOVAKIS
 Set Theory

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