 By P. M. Cohn

ISBN-10: 0471101699

ISBN-13: 9780471101697

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Additional resources for Algebra. Volume 1. Second Edition

Example text

7. Let a and b be objects. Then the set {afb} is called the unordered pair of a and b. In particular, for any object a, {a,a} is a set. By virtue of the axiom of extension, {aya} — {a}. Therefore {a} is a set containing a as its only element and we call this set the singleton of a. T h e axiom of pairing, useful though it is, still does not enable us to construct sets containing more than two elements. But we are not going to postulate similar axioms for three, four, . . elements since a single axiom, which we shall postulate later in Section G, will render this step unnecessary.

2. Two sets A and B are equal if and only if A is a subset of B and B is a subset of A. Because of this theorem, the proof of the statement that two sets A and B are equal is split into two parts; first prove that A c B and then prove that B <= A. 26 SETS [Chap. 2 The inclusion has the following properties: (a) For every set A, A xeA. To prove statement (b), we have by hypothesis, for all xy xeC => xeB, and xeB => xeA.

Because of this theorem, the proof of the statement that two sets A and B are equal is split into two parts; first prove that A c B and then prove that B <= A. 26 SETS [Chap. 2 The inclusion has the following properties: (a) For every set A, A xeA. To prove statement (b), we have by hypothesis, for all xy xeC => xeB, and xeB => xeA. e. C a A. This proves (b). Statement (a) means that every set is a subset of itself.