# New PDF release: Abstract Set Theory

By Abraham A. Fraenkel

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**Sample text**

E. any element of S,; then n is not contained in the set and therefore not in the meet. o different natural numbers. In other words : I n A srquenrcl - in corit,rast, to a set, cf. p. 17 - the same element may appmr several times. For the connection between tkle concepts of set and of sequence, see 4. g. (,TI, s,, s,,. ). Since sets are tlcnoted by curly brackets { }, no confusion will arise between a set and it sequence, if eit’her is denoted by its members. It is necessary to add “inner” because we shall yet require another 3, kind of mult iplication of sets (S 6 ) in respect of which we speak of the “outer” product.

The sum of denumerably many different sets, each of which is denumerable or finite, is again a denumerable set 3). + In view of the geometrical realization of real numbers, given in the example f ) of 3 1 (p. 12), one may express the denumerability of the set of all rationals in a geometric form. The line of numbers used in that example furnishes a representation of the set of all real numbers on the set of points on a straight line. Using this l) a) a) See Faber 1, Oglobin 1, Boehm 2, Godfrey 1, Johnston 1.

S,. The reader will easily transfer this argument t o the direction from right t o left and prove the second distributive law in quite a similar way. It should be stressed that for the operations called in arithmetic (and also in 9 6) addition and multiplication, only the first distributive law a ( b + c) = a . b + a c holds true, and not the second. Finally, a definition of rather limited significance : - DEFINITION IV. If So is a subset l ) of the set S,the set of all those elements of S that do not belong to So is denoted by S - So (difference of S and So).

### Abstract Set Theory by Abraham A. Fraenkel

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