Bridge to Abstract Math. Mathematical Proof and Structures by Ronald P. Morash

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 a company beginning in units, common sense, and mathematical facts tools. the second one variation features a smoother transition from the suggestions of good judgment to real use of those strategies in proving theorems; extra purposes; numerous essays approximately sought after mathematicians and their paintings; and the addition of routines for pupil writing.

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Example text

As stated earlier, we never list an object more than once as an element of a set. 2 OPERATIONS ON SETS 17 solution to Example 1 lists numbers in increasing order, this is not necessary. Observe also that the sets that result from the operation of union tend to be relatively large, whereas those obtained through intersection are relatively small. 1. Finally, our previous example introduces the use of parentheses, as in the algebra of numbers, to set priorities when an expression contains more than one instance of a set theoretic operation.

The latter are important, for example, in abstract algebra, with the proof of the famous theorem of Lagrange from group theory a case in point. The material in this article may be familiar to readers who have studied elementary probability. 5 COUNTING PROPERTIES OF FINITE SETS (OPTIONAL) 43 We emphasize that the content of this article is restricted to finite sets.

Solution Let W = (- oo,3), X W A X and W A Y. EXAMPLE 8 = (- 3,5], and Y = [4, oo). Compute W A X =(-a, -33 u [3,5] and W A Y =(-c0,3) u [4, a). You should calculate X A Y. Solution ORDERED PAIRS AND THE CARTESIAN PRODUCT The sixth and final operation on sets to be introduced in this article, cartesian product of sets, differs from the preceding five in a subtle but important respect. If U were the universal set for sets A and B, it would again be the universal set for A n B, A u B, A', A - B, and A A B.

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