Unformatted text preview: , the order on N is the usual order, and the order on S is ⊂ . 5. A relation ≺ on a set S is called irreﬂexive if for every x ∈ S , x 6≺ x . We call a relation a strict partial order if it is irreﬂexive and transitive. Prove that a strict partial order is antisymmetric. 6. Let C [0 , 1] be the set of continuous realvalued functions on [0 , 1]. Deﬁne a relation ≺ on C [0 , 1] by f ≺ g if and only if Z 1 ( gf )( x ) dx > . Show that ≺ is a strict partial order. 7. Let ≺ be a strict partial order on S . Deﬁne a relation ± on S by x ± y if and only if x ≺ y or x = y . Show that ± is a partial order on S....
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 Spring '10
 Math, Total order, strict partial order

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