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 winning textual content presents a company origin in units, common sense, and mathematical facts equipment. the second one version incorporates a smoother transition from the techniques of common sense to real use of those innovations in proving theorems; extra functions; numerous essays approximately popular mathematicians and their paintings; and the addition of workouts for scholar writing.

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

Sample text

The solution set to each of the following inequalities can be expressed as the union of intervals. Find them in each case: (a) 13x - 231 2 4 (c) (x - 5)/(5 - X)< 0 *(e) (3x2 - 27)/[(x - 3)(x + 3)] > 0 *(b) (d) 2x2-4x-9620 14x - 171 > 0 11. Solve simultaneously the pairs of inequalities: 1x1 2 1 and x2 - 4 5 0 ) 4 x + 8 ) < 12 and x 2 + 6 x + 8 > 0 *(a) (b) 12. Let U be the set of all functions having R as domain and range a subset of R. Let: A = { f ( f is continuous at each x E R) 1 B = { f f is differentiableat each x E R) I D = { f f is a quadratic polynomial) F = { f 1 f is a linear polynomial) List all subset relationships between pairs of these six sets.

The upshot of the associative laws for union and intersection is that the union of three or more sets consists of all the objects in any of the sets, grouped together within one set, while the intersection of three or more sets consists of the objects common to all the sets. Commutativity says that, in computing unions and intersections of two sets, the order in which the two sets are listed is irrelevant. DISTRIBUTIVITY Distributivity, familiar as a property of the real numbers, has its analogy in set theory.

Solution The result of Example 3 supports our conjecture. Do you think that (X n Y) u Z = X n (Y u Z) holds for any three sets X, Y, and Z? Test this conjecture further by using the three sets A, B and C of Example 2. 8. In this case the regions in the two Venn diagrams do not correspond. " confirm your results for the sets ( A n B) u C ( = {2,3,4,5,9, 10)) and A n (B u C ) ( = (2,3,5)), calculated from Example 2. Thus the conjecture of the preceding paragraph is false. This illustrates the danger of believing a general conclusion too readily, based on only a few examples, and especially, without seeing a proof.