1. Assume that 3' :RE i R3 is an isnmetry of the plane. Shaw that: {a} if f xes twn pnints n. and h nf the plane. then f is a reectinn relative to the line determined by n. and b; (h) if f xes a unique pnint a, then f is a rntatinn arnund a; {e} if 1" does net x any pnint, then f is a translatinn, fnnwed pnssihly by a retatinn andfer a reeetinn. 5. Shaw that the felinwing lets nf transfnrinatinns are grnups: {a} The nrthngenal transfnrmatiens and prnper rntatinns f : R\" > R\". {h} The isnmetries f : R\" i- R\". {e} The symmetries of a gure [3! C IR\". 5. A set X is said to he INFINITE 5 if there exists an injectiye map EF : X -> X which is net surjeetiye. Shnw that N is innite. . Shaw that, if there exists a bijeetinn iII' : X -+ 1', then X is innite if and only if 1" is innite. 1 Shaw that, if 1' C X and Y is innite, then X is alsn innite. In particular, shew that E is innite. 2. Show that, if A is a ring with identity 1 % and there exists in A an element 1' such that 2 = 1, then A earmet he nrdered