Download PDF by Abraham A. Fraenkel: Abstract Set Theory

By Abraham A. Fraenkel

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Example text

The same thing applies to inverting a fiinction. CH. I, 9 21 THE FUNDAMENTAL CONCEPTS. F I N I T E AND I N F I N I T E 33 set S while the values of the dependent variable y belong to T ; these values, however, need neither exhaust all elements of T nor be different from each other. e. to comprehend z as a function of the same kind; possibly different values of s will be related to one value of y. Take for instance the temperature y as a function of the time s (at a certain place). At any moment there is a definite temperature, but to a certain temperature different times may be related (possibly one in the morning and one in the evening) while temperatures exist to which no time corresponds, because they are never reached.

CONCEPT O F CARDINAL NUMBER [CH. (Sl+S3). Proof. According to the definition of equality, one has to show for either of these relations that any object contained in the lefthand set is contained in the right-hand set, and vice versa. Now, an element of S, ( S , + S,)belongs to S, as well as to S,+ S3; in other words, it belongs to 8, and to S,or S,. Therefore, it belongs to 8, and S,, or to S, and X,. 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.

E. e. t’he banana corresponding to 1 of T ) ;and so forth for each apple. I n other words: one omits the second line of the scheme, containing T , arid relates each banana to the apple located above it. - This procedure is a kind of inversion of the process mentioned on p. e. by enumerating their elements. V’e may thus formulate our results: THEOREM 3. The equivalence of sets (definition V) is a reflexive, symmetrical, and transitive relation; that is to say: S S ; S T implies T S ; X I’ and T W together imply S W .

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Abstract Set Theory by Abraham A. Fraenkel


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