By Keshawa Prasad Gupta

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Or :x n+a=a Write B={blt b2 , ba.... , bn1 A, being an enumerable set, is expressible in the form of a sequence. So we can write / A={an~l' an+2. anH. an+, ... } Define a map f: N-B U -A by the formula /(r)=l br ~f 1 <;; r <;; n 1Qr If r ~ n Evidently r is one-one onto. Hence N "'" B U A. =>_1 N /=/ B u A/=> a=n+a. Proved. 3S CC::>UNTAB(LITY OF ,SETS Theorem 10. cardinal number. Prove n+«=~ 'V n E N, at being any infinite Or If a finite set of elements is added to an infinite set, the power of the set is unaffected.

Prove n+«=~ 'V n E N, at being any infinite Or If a finite set of elements is added to an infinite set, the power of the set is unaffected. Proof. Let A be any infinite set with power IX. Let C be any finite set with cardinal number n. Let A n C=rP. We have to prove that 1A U C 1=1 A I. For proving this we prove that cx+n=lX. t. I B I==a. Write A-B=P. Then AU C=B U PuC or A U C=B U CUP. .. (1) B U C is enumerable, being a finite union of a finite set and an enumerable set. " N. , N, => N '" B.

3) ... (4) (x) C A, since P (A) is the family of all subsets of A. Also B C A, according to (4). Thus, Be A,/(x) C A for any x E A. This . 3 CIt E A. )=B. CIt e: A. => 11. e: B or tx ft. B. Consider the case in which 11. e: B. .. (5) Now, /It e: A. ell e: B => ell ft. ), according to (4) . • ell ft. B, Contrary to (5). : I (tx)=B. :. The possibility Ie e: B is ruled out. Consider the second possibility in which tx fF. B. (6) Now, • e: A. => ell ft. B ... : I (II)-B. . II e: B according to (4).

### Measure Theory (Including Banach Space, Hilbert Space, Spectral Theory) by Keshawa Prasad Gupta

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