 # question 1 Let A and B be two mon-empty sets and let f A - B be a ...

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question 1 Let A and B be two mon-empty sets and let f A - B be a function. Define a relation a ~ 6 for a. b € A by a 6if and only if f(a) = f(b) a) show that a ~b is an equivalence relation b) Describe corresponding equivalency classes c) If f is injective, how many representatives do ench equivalency class have? question 2 and h(x) f(x) = f(x) a) Show that h is a contractual image on R b) Given an arbitrary pointp € R and let P1 = h(p),P2 = h(p2).Ps = h(p2) etc, ie Pn = hT.show that Pa-> them question 3 Assume that the functionf (0,00) - (0, bo) is continuous and decreasing. Show that limit lim (Even-Trane) exists. (Tip: that monotonous and limited numbers of resal numbers are convergent, may be used without proof.) 1

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