# Abstract Set Theory by Abraham A. Fraenkel

By Abraham A. Fraenkel

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Example text

Using this l) a) a) See Faber 1, Oglobin 1, Boehm 2, Godfrey 1, Johnston 1. I n view of theorem 1, it is superfluous t o define precisely the subsets. It is easy t o see that the sum cannot be finite.

Equivalence has different meanings in different branches of mathematics. However, all these meanings share the three properties expressed in theorem 3. Therefore, equivalence is sometimes used in a more general sense, meaning any relation with two arguments having those three properties I). Also the equality of sets ( 1 1 . e. a relation of - - _____ -- - - - - These 1)ropertics are not irzdependepkt. For S T implies T --S by I) the symmetry and these relations imply S S by the transitivity. Accordingly, if there itre at least two eq~iivalentobjects, the reflexivity is a logical conscqumco of the symmetry and the transitivity.

4) A relation is defined only after the domains of variability for its arguments x, y etc. have been fixed. Try to comprehend this in view of the instance “x is a brother of y”. (It depends on the determination of the domain of variability for y whether this relation is symmetrical or not. ) 5) How complicated the connection between reflexivity, symmetry and transitivity of relations is in general (cf. the footnote on p. 36), one may gather from the following example: The relation between two arguments ‘‘x and y are prime numbers”, defined for integers x and y, certainly is symmetrical and also transitive.