By Ronald P. Morash

Meant to bridge the distance among the traditional calculus series and extra summary upper-division arithmetic classes, this profitable textual content offers an organization starting place in units, common sense, and mathematical facts equipment. the second one version encompasses a smoother transition from the strategies of good judgment to real use of those strategies in proving theorems; extra functions; numerous essays approximately well known mathematicians and their paintings; and the addition of routines for pupil writing.

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**Extra info for Bridge to Abstract Math. Mathematical Proof and Structures**

**Example text**

This equation resembles the distributive law for real numbers, a (b + c) = a b + a . c, if we substitute intersection for "times" and union for "plus," and so seems like a plausible candidate on that basis as well. h 34 SETS Chapter 1 To get further evidence, you should test this equation by using sets A, B, and C of Example 2 and also by drawing Venn diagrams. The results of this work should support (although of course not prove) the conjecture that this equation, which asserts that "intersection distributes over union," holds for all sets X, Y, and 2.

The elements of A x B are ordered pairs of elements from A and B, and thus are not ordinarily members of the universal set for A and B. Ordered pairs resemble notationally two-element sets but differ in two important respects. Represented by the symbol (a, b), the "ordered pair a comma b" differs from the set (a, b} in that the order in which the elements are listed makes a difference. Specifically, the ordered pairs (a, b) and (b, a) are different, or unequal, unless a = b, based on Definition 6.

List ten such elements. (c) Can you give an example of a finite set X such that #(X)is infinite? s . "10. Suppose U were a truly universal set; that is, U contains all objects. Then, in particular, U would contain itself as an element, that is, U E U. ) Now consider the "set" A of all sets that are not elements of themselves; that is, A = {YIY q! Y). Discuss whether A E A or A $ A. 2 Operations on Sets As stated earlier, sets like Z or R or (1,2, 31, consisting of numbers, are of greater mathematical interest than most other sets, such as the set of all names in a telephone book.