Abstract Set Theory - download pdf or read online

By Abraham A. Fraenkel

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S $ and S’ are again exclusibe. Now, we construct a representation between the Yets S - S* S’ and So = S: S’ S + + + b~ a tloublr step : a ) any element of S belonging to S’ shall be related t o itself; b ) any elerneiit of S belonging to S*, and so being of the form sk, shall be related to the element s ? ; + ~of S,*(and of So). This rule evidently creates a one-to-one correspondence between the elements of S* and S:, being completely analogous t o the reprerentation between N and N‘ considcred on page 39.

The proof is trivial, since the oquiralence is shown by the identical representation which relates every element to itself. This property is expressed as follows: equivalence is a reflexive relation. The null-set, too, is called equivalent to itself. Secondly, if 8 i s equivalent to T , T is also equivalent to S ; in symbols: 8 T implies T X. , with the relation “x is larger than y” (and any other relation of order) which is even incompatible with “y is larger than x”. This property of equivalence is the immediate consequence of the biuniqueness of the correspondence used for the definition of equivalence.

S $ and S’ are again exclusibe. Now, we construct a representation between the Yets S - S* S’ and So = S: S’ S + + + b~ a tloublr step : a ) any element of S belonging to S’ shall be related t o itself; b ) any elerneiit of S belonging to S*, and so being of the form sk, shall be related to the element s ? ; + ~of S,*(and of So). This rule evidently creates a one-to-one correspondence between the elements of S* and S:, being completely analogous t o the reprerentation between N and N‘ considcred on page 39.

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