By Ronald P. Morash

Meant to bridge the space among the normal calculus series and extra summary upper-division arithmetic classes, this winning textual content presents a company starting place in units, good judgment, and mathematical evidence equipment. the second one version contains a smoother transition from the strategies of good judgment to genuine use of those recommendations in proving theorems; extra functions; numerous essays approximately popular mathematicians and their paintings; and the addition of routines for scholar writing.

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The intersection of A and D in Example 2 provides another justification for the existence of an empty set since A and D have no elements in common. Pairs of sets such as A and D, having no elements in common, are said to be disjoint. You should perform other calculations involving the sets in Examples 1 and 2, for instance, B n F. What about the intersection of the empty set with another set? In particular, what is A n Finally, does our calculation of E n F and E u F suggest any possible theorem about intervals?

The operations of union and intersection are called binary operations because they are applied to two sets to make a third set. Let A = {1,3,5,7,9), B = {1,4,7, 10,13,16), and C = (-5, -3, -1, 1,3,5). Calculate A n B, A u B , A n C, B n C, and B u ( A n C). EXAMPLE 1 A n B = (1, 7) since these two objects are common to both sets and are the only such objects. A u B = (1, 3,4, 5,7,9, 10, 13, 16) since this set results from "gathering into one set" the elements of A and B. Similarly, A n C = (1, 3, 5) and B n C = (1 ).

X U X' = U 23. X n X'= @ 24. V = @ 25. 4 THEOREMS OF SET THEORY 41 FACT 4 The following distributive laws can be proved to be theorems of set theory. For all sets XI Y, and Z in any universal set U: 26. X u (Y n Z)= (X u Y) n (X u Z) 27. X n (Y u Z) = (X n Y) u (X n Z) 28. X n (Y A Z)= (X n Y) A (X n z ) (union over intersection) (intersection over union) (intersection over symmetric difference) FACT 5 The following basic properties of set difference can be proved to be theorems of set theory. For all sets X, Y, and Z in any universal set U: 29.

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