By Felix Hausdorff

This paintings is a translation into English of the 3rd version of the vintage German language paintings Mengenlehre by way of Felix Hausdorff released in 1937. From the Preface (1937): ``The current e-book has as its objective an exposition of crucial theorems of the idea of units, besides whole proofs, in order that the reader aren't locate it essential to cross open air this e-book for supplementary info whereas, nonetheless, the ebook should still permit him to adopt a extra unique examine of the voluminous literature at the topic. The ebook doesn't presuppose any mathematical wisdom past the differential and crucial calculus, however it does require a definite adulthood in summary reasoning; certified university seniors and primary yr graduate scholars should not have any trouble in making the cloth their very own ... The mathematician will ... locate during this booklet a few issues that would be new to him, at the least as regards formal presentation and, specifically, as regards the strengthening of theorems, the simplification of proofs, and the elimination of pointless hypotheses.''

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0 0 0 0 0 0 . . 0 0 • • • an-2 0 0 0 -1 x 3 0 0 -1 0 0 0 an-1 2 1 0 0 0 0 x 0 -1 x 0 0 an CH. 2. EVALUATION OF DETERMINANTS *190. Compute the difference f(x +1) —f(x), where 1 1 f (x)= 1 0 2 3 0 0 3 1n 1 n+1 0 0 0 ... 0 ... 0 0 x2 x3 . . C2 C3 C „2+1 C3„+1 . . xn `-'n+1 ,n+1 Compute the determinants: *191. x a1 a2 al x a2 al a2 an-1 1 a„_, 1 a„_, 1 al a2 a3 . . x al a2 a3 . . an 193. *194. *195. x —a —a a x —a —a —a —al 0 0 a, a2 0 0 1 0 1 a, 0 0 a2 0 0 1 0 1 *192. 1 1 a ... a ... x...

N- 1 . . n-2 . • . n-3 4 3 2 1 n-1 n-2 n-3 . . ao a1 a2 an_1 a„ b1 0 0 ... — b, 0 b2 0 ... 1 0 0 —b1 b, . 0 0 0 0 0 0 0 0 0 ... 0 ... b„ 0 —bn-1 *204. 0 a a2 0 . 0 1 2a+b (a+b)2 0 1 2a + 3b (a + 2b)2 . . 0 0 0 0 0 x y 0 0 207. 0 . . 2a + (2n — 1) b (a + nb)2 1 2a+(2n+1)b 0... 0 x 1 + x2 Y1 + xiY2 . 1+ xi y„ + X2 Y2 ... 1+ x2 y„ 1+ x„yi 1 +xnY2 ... 1+ x„y„ 1 + x1Y1 1 al —b1al—b2 a1— b„ a, — b1a2 — b2 • • • a2— b. a„—b1 a n —b2 • • • a„—b„ *208. 0 0 0 *206. *205. 0 y 0 0 0 0 a1-Fx2 • •• a2 -1-•x1 1+a2 -Fx2 an + xi a„+ 1+an + x„ 38 PART I.

Sin ny n sin ny n cos ny cos ny + 1) (I D sin(n+ 1)cp(n+ 1)sin(n+ 1)y (n cos(n+ 1)cp (n + 1) cos cos (n + 2) y (n + 2) cos (n + 2) y sin (n + 2) y + 2) sin (n + 2) y cos (n + 3) y +3) cos (n + 3) y sin (n + 3) y (n + 3) sin (n + 3) cp *300. *299. a1 a2 . . an _1 ao 1 . 1 1 1 n-1 e 2 an _1 a0 • • an-2 e 1 E 1 o 1 en-1 e2n-2 en-1)2 e2 (n-1) where s = cos i sin 27` — • a1 a2 a3 . . a0 cyclic determinant). 53 54 PART I. PROBLEMS 301. Apply the result of Problem 300 to the determinant x it z z y u z y x 302.