Unformatted text preview: X ⊆ Y with Y a ﬁnite set. Use the previous problem to show that (a) X = Y if and only if  X  =  Y  . (b) X ± Y if and only if  X  <  Y  . (6) Suppose that X and Y are nonempty ﬁnite sets of cardinality n . Prove that a map X → Y is a surjection if and only if it is an injection. (7) Let X and Y be ﬁnite sets (possibly empty). Prove that  X × Y  =  X  ×  Y  . (8) Suppose that { 1 ,...,n } → X is a surjection. Prove that  X  ≤ n . (Hint: Do induction on n .) (9) Let f : X → Y be a function between metric spaces ( X,d ), ( Y,f ). Using the deﬁnition of continuity from class, prove that f is continuous if and only if the inverse image of every open set is open, i.e. for all open sets U ⊂ Y , f1 U is an open set in X . 1...
View
Full Document
 Spring '09
 DANBERWICK
 Math, Division, Empty set, Natural number, Topological space, Countable set

Click to edit the document details